ELECTROTONIC COUPLING REDUCES ACTION POTENTIAL DURATION GRADIENTS IN THE VENTRICLE: A SIMULATION STUDY

M.-C. Trudel¹, R. M. Gulrajani¹, L. J. Leon²
¹Institute of Biomedical Engineering, Université de Montréal,
P.O. Box 6128, Station Centre-ville, Montreal, Quebec, CANADA H3C 3J7
²Department of Electrical and Computer Engineering, University of Calgary,
2500 University Drive N.W., Calgary, Alberta, CANADA, T2N 1N4

Abstract: It has been hypothesized that in situ electrotonic coupling in the myocardium can diminish the gradients of action potential duration across the ventricular wall. Using a computer heart model that incorporates membrane-based representations for the model cells, we verify that this is indeed the case.

INTRODUCTION

Present-day computer heart models have started to incorporate membrane-based representations for the model cells. Huiskamp [1] recently described one such model using the modified Beeler-Reuter membrane model, and we [2] have more recently described another employing a modified Luo-Rudy phase 1 membrane model. Propagation in these models is achieved in natural fashion, with electrotonic coupling between adjacent model cells bringing the individual cells to threshold and triggering an action potential. Such models enable one to easily observe the subtle effects of electrotonic coupling on the action potential waveform.

METHODS

The anatomy of our heart model is identical to one published earlier by Lorange and Gulrajani [3]. Now, however, instead of exciting the model by simply stipulating conduction velocities along and across cardiac fibers, the more correct reaction-diffusion equation is used. To enable correct numerical integration of this equation, it was necessary to augment the spatial resolution of the Lorange-Gulrajani model from 1 mm to 0.25 mm. On account of the large number of model points that resulted (approximately 12 million), only the ventricles of the model were excited. Also, parallel processing was used in the integration of the reaction-diffusion equation, employing 16 processors of a Silicon Graphics Origin 2000 computer.

The reaction-diffusion equation is derived assuming a bidomain representation for the myocardium, in which the intracellular and interstitial domains are characterized by the equations

(1)

(2)

respectively. Here and are the intracellular and interstitial potentials, respectively, and the intracellular and interstitial conductivity tensors, respectively, the surface to volume ratio of the cardiac cells, the transmembrane current coupling the two domains, and the intracellular stimulation current used to start excitation at the selected endocardium points. The transmembrane current passes from the intracellular to the interstitial space and is given by the sum of the capacitive and ionic currents,

(3)

where is the specific membrane capacitance, and is the transmembrane potential. As mentioned, the ionic current was represented by the Luo-Rudy phase 1 membrane model [4], with its action potential duration modified, however, to match those of human ventricular cells. The combination of equations (1), (2) and (3) together with the approximation of equal anisotropy in the intracellular and interstitial spaces results in the reaction-diffusion equation

(4)

Finite-differences were used to solve Equation (4) over the irregular ventricular domain. Since fiber directions rotate counterclockwise from epicardium to endocardium in our model, the diffusion tensor varies from point to point. Interstitial conductivities along and across cardiac fibers were taken to be 5.32 and 1.33 mS/cm, respectively, for an anisotropy ratio of 4. The parameter was taken as 0.5 so that intracellular conductivities were half the above interstitial values. The value of was selected to control the total excitation time, which was approximately 100 ms for complete activation of the ventricles.

Also incorporated into our heart model were the M cells found in the mid-ventricular wall [5]. These M cells were also simulated via modified Luo-Rudy equations, and had an intrinsic duration 80 ms longer than those of endocardial or epicardial cells. All modifications entailed decreasing the time constants , responsible for the slow inward current in the Luo-Rudy model, by a factor of 0.5 for endocardial and epicardial cells and a factor of 0.625 for M cells. In addition, the potassium conductance , responsible for the time-independent potassium current , was diminished for the M cells. These changes produce a transmural gradient of during repolarization that contributes to the upright T waves noted in the normal electrocardiogram (ECG).

Figure 1: Action potential waveforms for endocardial or epicardial cells (duration 239 ms) and for M cells (duration 321 ms). Figure 2: Endocardial, epicardial and M-cell waveforms during normal activation, with respective durations of 258, 243 and 300 ms.

Simulation of the ECG was done by using the spatial gradient of the transmembrane potential determined from Equation (4) to calculate elemental current dipoles at each model point. The elemental dipoles were combined by region to result in 58 time-varying current dipoles, that, in conjunction with a human inhomogeneous torso model, were then used to calculate the torso surface potentials using a standard integral equation approach.

RESULTS

The ECG generated by the heart model corresponded well to those observed in normals [2]. Here we focus, however, on the effect of electrotonic coupling on the ventricular action potential. Figure 1 shows the action potential waveforms of the endocardial or epicardial cells, and of the M cells, as determined by the modified Luo-Rudy phase 1 membrane model. The intrinsic 82 ms difference in action potential duration is evident. Figure 2 shows the endocardial, M-cell and epicardial waveforms across the left ventricular wall during normal activation of the heart model, in other words with electrotonic coupling present. The 82 ms difference in action potential duration has been whittled down to 42-57 ms, largely due to an increase in endocardial and epicardial action potential durations.

DISCUSSION

We clearly demonstrate the reduction in intrinsic action potential durations across the ventricular wall due to electrotonic coupling. Anyukhovsky et al. [6] noted that the difference in action potential durations between M cells and endocardial or epicardial cells was greater in vitro than in situ, and suggested that this was due to electrotonic coupling. Verification of this electrotonic coupling hypothesis is difficult experimentally, but as evidenced by the above results, is easy to accomplish via computer simulation.

Acknowledgments: Work supported by the Natural Sciences and Engineering Research Council of Canada, and by FCAR, Québec. M.-C. Trudel was supported in part by a Scholarship from la Fondation J. A. de Sève.

REFERENCES

[1] G. Huiskamp, “Simulation of depolarization in a membrane-equations-based model of the anisotropic ventricle,” IEEE Trans. Biomed. Eng., vol. 45, pp. 847-855, 1998.

[2] M.-C. Trudel, R.M. Gulrajani, L.J. Leon, “Simulation of propagation in a realistic-geometry computer heart model with parallel processing,” CDROM Proceedings, 23rd Annual International Conference IEEE Eng. Med. Biol. Soc., 2001.

[3] M. Lorange and R.M. Gulrajani, “A computer heart model incorporating myocardial anisotropy. I. Model construction and simulation of normal activation,” J. Electrocardiol., vol. 26, pp. 245-261, 1993.

[4] C. Luo and Y. Rudy, “A model of the ventricular cardiac action potential. Depolarization, repolarization and their interaction,” Circ. Res., vol. 68, pp. 1501-1526, 1991.

[5] S. Sicouri and C. Antzelevitch, “A subpopulation of cells with unique electrophysiological properties in the deep subepicardium of the canine ventricle,” Circ. Res., vol. 68, pp. 1729-1741, 1991.

[6] E.P. Anyukhovsky, E.A. Sosunov, R.Z. Gainulin, and M.R. Rosen, “The controversial M cell,” J. Cardiovasc. Electrophysiol., vol. 10, pp. 244-260, 1999.