Introduction
A simplified electric double layer model of myocardial electrical activity is widely used in cardiac electrophysiology (see, for instance, [Barr, R.C., Van Oosterom, A., 2010]). This model represents the cardiac electrical potential as a potential of a double layer given on the myocardium surface. Such a representation is a useful tool for studying the electrical activity of the heart and non-invasive cardiac mapping. However, the derivation of the relations of this model, reported in the available literature, is not mathematically rigorous, and some formulas need to be clarified.
Methods
We investigated the problem theoretically using some results for elliptic PDE [Shlapunov A., Tarkhanov, 1995] and the bidomain model [Kalinin V, Shlapunov A, Ushenin K, 2022].
Results
We obtained a rigorous derivation of the double layer model based on the bidomain model. We showed that for the case of isotropic homogeneous electrical conductivity, the double layer density is proportional to the transmembrane potential only if the sum of the values of extracellular and intracellular electrical conductivity is equal to the value of extra-cardiac electrical conductivity. We also demonstrated that there is no double layer, the potential of which simultaneously coincides with the potential on the torso surface and on the myocardium surface, if the torso is surrounded by a non-conductive medium (air). In this regard, we presented refined formulas for computing the electric potential on the torso surface by the density of the double layer, the potential of which coincides with the potential on the cardiac surface. Thus, we obtained the correct form of the transfer matrix for the inverse problem of electrocardiography in terms of the double layer or transmembrane potential.
Conclusion
The results of this work can be used in computational electrophysiology of the heart.