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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 » Results

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

Figure 1 shows Vm(x,y) for isotropic tissue with λx = λy = 1 mm. The electrode is located above the center of the plot. The color indicates the value of the transmembrane potential, with yellow indicating depolarization, purple rest, and blue hyperpolarization. The tissue is depolarized by 30 mV directly under the electrode. The region of depolarization is surrounded by halo of weak, diffuse hyperpolarization with a peak value of -0.6 mV, which is difficult to distinguish from resting tissue using the color scale in Fig. 1. Isocontours of the transmembrane potential are concentric circles, implying that Vm is independent of direction.

Figure 2 shows Vm(x,y) for λx = 1 mm and λy = 0 mm. The x direction (fiber axis) is horizontal. Like in Fig. 1, the tissue is depolarized under the electrode (20 mV). However, hyperpolarization exists to the left and right of the electrode, with a peak hyperpolarization of -3.8 mV. This simulation corresponds to no coupling of cells in the transverse direction, as would be appropriate for a nerve.

Figure 3 shows Vm when λx = 1 mm and λy = 0.4 mm. The distribution of Vm is qualitatively similar to that in Fig. 2, with depolarization under the electrode (23 mV) and hyperpolarization on each side of the electrode along the fiber direction (-2.5 mV). This simulation uses parameters typical of normal cardiac tissue [14].

The transmembrane potential depends sensitively on the height of the electrode above the plane of the tissue sheet, d. Figure 4 shows the peak depolarization as a function of d for the case of normal cardiac tissue. The dashed line has a slope of negative two on this log-log plot, and corresponds to a 1/d2 fall off of the depolarization. This figure explains why the stimulation threshold changes when an electrode is moved away from the monolayer.

To test our numerical accuracy, we decrease the space step size to 0.05 mm and increase the number of nodes to 400 × 400 while keeping the tissue size fixed, and find that the maximum and minimum values of Vm vary by about 1%. In another simulation, we keep the space step constant at 0.2 mm and increase the number of nodes to 400 × 400, thereby increasing the size of the tissue sheet to 80 mm × 80 mm. The maximum and minimum values of Vm vary by 3%.


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