Role of the Second Spatial Derivative of the Extracellular Potential in Field Stimulation

Initial tests were performed with nonsimultaneous electrical and optical mapping.15 Cable theory predicts transmembrane current is proportional to the second spatial derivative of the extracellular potential (i.e., Laplacian component).16'17 A Laplacian component differs from the extracellular potential gradient (first spatial derivative of extracellular potential), which has been used to quantify stimulatory strength and thresholds of the shock-induced electric field in tissue.18'19

To produce Laplacian components in two dimensions on the heart surface, electrical stimulation was applied with metallic wire or mesh electrodes to produce either nonuniform or approximately uniform electric fields on the epicardium. Optical mapping in a region between the electrodes indicated the changes in transmembrane potential produced by the electric fields. In separate measurements, extracellular potentials were mapped with a roving linear electrode array while the stimuli were repeated for each position of the array. Results were compared with bidomain models that incorporated the heart's epicardial fiber structure.

Results showed that the location of the detectable Laplacian components of the extracellular potentials qualitatively match locations of changes in transmembrane potential. As illustrated in Fig. 1, regions undergoing positive changes in transmembrane potential exhibit a mostly negative Laplacian, while regions undergoing negative changes in transmembrane potential exhibit a positive Laplacian. The changes in transmembrane potential correspond to the Laplacian components more than they correspond to the potential gradient. Also the results demonstrated that fiber structure of the tissue distorts the extracellular electric field.

In a study to test the activating function theory,20'21 ITO was used to create an extracellular electric field containing an activating function.7 Half of the ITO film was etched in stirred acid solution for a brief time to thin the ITO and increase its sheet resistivity. Then leads were attached to the etched and nonetched ends of the ITO film as shown in Fig. 2. Current applied to the ITO produced a large potential gradient in the etched ITO region, while it produced a smaller potential gradient in the nonetched ITO. An activating function occurred at the boundary between the two regions where the potential gradient changed.

In experiments with rabbit hearts stained with di-4-ANEPPS, the ITO was placed on the heart so that the ventricular epicardial surface was exposed to the activating function. Optical mapping was performed in the underlying tissue. The excitation light and fluorescence passed through the ITO and glass plate. Results are illustrated in Fig. 3.

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Figure 1: Maps of the Laplacian of extracellular potential during electric field stimulation performed with a small electrode above the center of the map and a large electrode below the map to produce a nonuniform electric field. (a) Epicardial measurements (V/cm2) from a rabbit heart. Transmembrane potential changes (numbers and contour lines) are overlaid on a gray-scale Laplacian from the same heart. (b) Bidomain model results. The current strength was 56 mA in the heart and 44 mA in the model. (From Knisley et al., Biophysical Journal 77, 1404-1417, 1999, figure 12 © 1999 Biophysical Society, Reproduced with permission.)

Figure 1: Maps of the Laplacian of extracellular potential during electric field stimulation performed with a small electrode above the center of the map and a large electrode below the map to produce a nonuniform electric field. (a) Epicardial measurements (V/cm2) from a rabbit heart. Transmembrane potential changes (numbers and contour lines) are overlaid on a gray-scale Laplacian from the same heart. (b) Bidomain model results. The current strength was 56 mA in the heart and 44 mA in the model. (From Knisley et al., Biophysical Journal 77, 1404-1417, 1999, figure 12 © 1999 Biophysical Society, Reproduced with permission.)

Figure 2: Production of activating function (second spatial derivative of extracellular potential) with indium tin oxide (ITO). (a) Right half of conductive ITO on glass plate was etched in acid to decrease thickness of the ITO film and increase film resistance. Shock current was delivered at plate ends. (b) Current produced potential gradients along the plate measured with roving electrode. Potential gradient in etched half was 2.5 times that in nonetched half. Second spatial derivative of extracellular potential occurred in central region. (c) Plate was positioned on ventricles of hearts stained with di-4-ANEPPS. Laser scanner measured transmembrane potential changes in central region. (From Knisley, IEEE Transactions on Biomedical Engineering 47, 1114-1119, 2000, figure 1, © 2000, IEEE, Reproduced with permission.)

Figure 2: Production of activating function (second spatial derivative of extracellular potential) with indium tin oxide (ITO). (a) Right half of conductive ITO on glass plate was etched in acid to decrease thickness of the ITO film and increase film resistance. Shock current was delivered at plate ends. (b) Current produced potential gradients along the plate measured with roving electrode. Potential gradient in etched half was 2.5 times that in nonetched half. Second spatial derivative of extracellular potential occurred in central region. (c) Plate was positioned on ventricles of hearts stained with di-4-ANEPPS. Laser scanner measured transmembrane potential changes in central region. (From Knisley, IEEE Transactions on Biomedical Engineering 47, 1114-1119, 2000, figure 1, © 2000, IEEE, Reproduced with permission.)

Figure 3: Outcome of experimental trials with activating function in central region (dotted lines) produced by nonuniformly etched indium tin oxide (ITO) plate. Action potentials with and without the shock are overlaid. The d2Ve/dt2 was negative (left column, V3" < 0) or positive (right column, V3" > 0). When the positive lead was attached to the low-gradient end of the ITO (left column), negative transmembrane potential changes occurred during the shock both before and after the plate was rotated. When the positive lead was attached to the high-gradient end of the ITO (right), positive transmembrane potential changes occurred before and after rotation. (From Knisley, IEEE Transactions on Biomedical Engineering 47, 1114-1119, 2000, figures 3 and 4, © 2000, IEEE, Reproduced with permission.)

Figure 3: Outcome of experimental trials with activating function in central region (dotted lines) produced by nonuniformly etched indium tin oxide (ITO) plate. Action potentials with and without the shock are overlaid. The d2Ve/dt2 was negative (left column, V3" < 0) or positive (right column, V3" > 0). When the positive lead was attached to the low-gradient end of the ITO (left column), negative transmembrane potential changes occurred during the shock both before and after the plate was rotated. When the positive lead was attached to the high-gradient end of the ITO (right), positive transmembrane potential changes occurred before and after rotation. (From Knisley, IEEE Transactions on Biomedical Engineering 47, 1114-1119, 2000, figures 3 and 4, © 2000, IEEE, Reproduced with permission.)

Results show that contact with the ITO containing an activating function introduces a change in transmembrane potential that has the corresponding sign. Other tests with only a potential gradient showed that transmembrane potential changes are due to a variation in heart conductance between the base and apex, corresponding to increased heart width near the base. These results are consistent with the generalized activating function that contains a term for the extracellular potential gradient scaled by the change in tissue conductance, and a term for the change in the potential gradient.21

The sign of the second derivative of the extracellular potential matches the sign of the transmembrane potential change in the experiments shown in Figs. 2 and 3, whereas it is the opposite of the sign of the transmembrane potential change in Fig. 1. This is due to differences in experimental conditions in the two studies. The experiments in which the ITO produced an activating function are more applicable to the Rattay formulation.20 The ITO resistance is sufficiently low that the electric field produced by ITO is not greatly affected by the tissue. That differs from the conditions used for the study in Fig. 1, in which the extracellular electric field was affected by the tissue.15'20

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