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The goal of our study is to examine the effect of stimulating …


Biology Articles » Biomathematics » A mathematical model for electrical stimulation of a monolayer of cardiac cells » Discussion

Discussion
- A mathematical model for electrical stimulation of a monolayer of cardiac cells

Figure 3 is our primary result. The transmembrane potential distribution is qualitatively similar to that calculated in a two-dimensional sheet by Sepulveda et al. [4]. In particular, both calculations predict adjacent regions of depolarization and hyperpolarization near the stimulating electrode. The main difference between our calculation and theirs is the presence of a perfusing bath in our model. We therefore conclude that the qualitative distribution of transmembrane potential is insensitive to the perfusing bath. This result is important, because the adjacent regions of depolarization and hyperpolarization were responsible for break excitation and reentry induction. Experimenters should be able to study these phenomena using superfused cell monolayers.

Our results are quantitatively different than those of Sepulveda et al. [4]. For instance, the peak depolarization in their calculation was infinite because they used a point electrode embedded in the sheet of tissue. We do not predict an infinite depolarization because our electrode is a distance d from the tissue sheet. One advantage of our calculation over Sepulveda et al.'s is that our stimulus current is expressed in milliamps whereas theirs is in meters-milliamps, as required by their fully two-dimensional model.

Our simulations are similar to those by Latimer and Roth [15], in which an electrode in an adjacent perfusing bath stimulated a three-dimensional slab of cardiac tissue. The main difference between our calculation and theirs is that their three-dimensional geometry implied certain boundary conditions on the potential and potential gradient at the tissue surface. In a three-dimensional model, a potential gradient can exist in the z-direction. Because the intracellular space is sealed, such an electric field polarizes the tissue within a few length constants of the surface [16]. This boundary effect is not present in our model because the tissue sheet is two-dimensional and therefore cannot have a transmembrane potential gradient in the z-direction.

Our results in Fig. 2 are almost identical to the results found by Rattay when studying the stimulation of nerve axons. Rattay referred to the source term in his partial differential equation for Vm as the "activating function" [13]. Sobie et al. [17] have analyzed electrical stimulation of cardiac tissue using a similar concept. If we use this terminology, the expression on the right-hand-side of Eq. (5),

is the activating function.

One approximation that is implicit in our analysis is that the cell monolayer itself does not significantly perturb the extracellular potential distribution, so we can use Eq. (8) for Ve. Rattay used a similar approximation for his analysis of nerve stimulation [13]. If the monolayer is grown on glass or any other insulating substrate, as is commonly the case, then the substrate will perturb the extracellular potential. For a monolayer grown directly on the surface of the substrate, image analysis [18] implies that the extracellular potential experienced by the monolayer will still be equal to the expression given in Eq. (8), except for an additional factor of two. To prove this surprising result, consider the potential produced by two current sources in an unbounded bath, one at x = y = 0 and z = d, and the other at x = y = 0 and z = -d. Let both sources have the same strength and polarity. The resulting potential distribution will be a generalization of Eq. (8)

This potential distribution obeys Poisson's equation for z > 0, and has zero derivative in the z-direction at the surface z = 0. Because of the uniqueness of the solution to Poisson's equation, Eq. (10) must therefore be the potential produced by a point current source a distance d from an insulating surface. Throughout the bath, Eq. (10) predicts a different potential than does Eq. (8). This difference can be thought of as arising from secondary sources at the insulating boundary, which are correctly accounted for by the image method [18]. However, at the surface z = 0 Eqs. (8) and (10) give identical results, except for a factor of two. This useful result means that a monolayer grown on an insulating surface will experience the same activating function as a monolayer grown in an infinite bath, except that the activating function is twice as large.

One interesting conclusion of our calculation is that the magnitude of the induced transmembrane potential varies inversely with the conductivity of the perfusing bath. This conclusion is true when the stimulating electrode is attached to a current source, as we assumed in our calculation (in our model, the stimulus current I is specified independently of the bath conductivity, the distance d, or any other parameters). If the conductivity of the bath increases, the magnitude of the induced polarization decreases (voltage drops are smaller for a given current), but the spatial distribution of Vm is unchanged.

In conclusion, the transmembrane potential induced when stimulating a monolayer of cardiac cells is similar to the transmembrane potential distribution predicted by Sepulveda et al. [4] when analyzing stimulation of a two-dimensional sheet of cardiac tissue. The presence of a perfusing bath above the sheet changes the quantitative value of Vm, but has little effect on its qualitative spatial distribution. Therefore, cell monolayers represent a well-controlled system which experimentalists could use to study break stimulation and the induction of reentry.

Acknowledgements
This research was supported by the National Institute of Health (RO1 HL57207).

For more information please contact BJ Roth (http://www.Oakland.edu/~roth



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