Analytical validation of the BEM--application of the BEM to the electrocardiographic forward and inverse problem

The objective of this study is to analytically validate a boundary element (BE) formulation for the relationship between the transmembrane potential on the heart's surface and the potential on the body surface applying a concentric spherical test geometry. The relative difference (reldif) between the potential on the outer sphere of the test geometry computed analytically and numerically is determined by 3.59% for the coarse discretization (48 BEs) and by 0.46% in the case of the finer subdivision (192 BEs). In the inverse problem, the transmembrane potential on the inner sphere is estimated numerically from the electric potential on the outer sphere by using a minimum-norm least-square approach. The relative differences found are 20.2% when no measurement noise is added and 26.4% in the presence of 2% additional Gaussian noise. The BE formulation is also applied to real world data for solving the electrocardiographic inverse problem. A normal volunteer's inhomogeneous thorax (outer thorax surface, surfaces of the lungs, epicardial heart surface) is modelled by 424 BEs. The same inverse method is then applied in order to reconstruct the transmembrane potential on the epicardium from the measured body surface potential (BSP) data during normal ventricular depolarisation.