On the Correctness of the Transmembrane Potential Based Inverse Problem of ECG

On a Transmission Problem Related to Models of Electrocardiology

“…Let us illustrate this by one model of electrocardiology [1,2,5]. Denote by Ω B and Ω H threedimensional domains with piecewise smooth boundaries with ∂Ω B and ∂Ω H corresponding to a body and a heart (see Fig.…”

“…Since matrices M i and M e are positively defined and not degenerate, the problems (2), (3) can be studied in the framework of the theory of boundary (maybe ill-posed) problems for elliptic formally self-adjoint equations, see [1,2,5]. Moreover, notice that the problems above may be regarded as transmission problems for elliptic equations with discontinuous coefficients describing solutions in different domains of a continuum with the help of additional conditions on separating surfaces, see, for example, [6,7].…”

“…Under assumptions of Theorem 3.1 the rest of the scheme of solving the problem (5), (6), (11) differs little from the standard one, see [5]. Namely, first we introduce a function h(x) such that h(x) = λ 2 u i + u e , where x ∈ Ω H .…”

“…Then by the uniqueness theorem for a Cauchy problem for elliptic operators (see, for example, [3, Theorem 10.3.5]), u b ≡ 0 in Ω. Now by the trace theorem for Sobolev spaces and by equations from(5) we see that u e ≡ 0 нon ∂Ω H and ν Ae (u e ) ≡ 0 on ∂Ω H . However, since the system of boundary operators {I, ν Ae } is a first order Dirichlet system on ∂Ω H , it follows from the theorem on spectral synthesis (see[10]) that u e ∈ [H 2 0 (Ω H )] k .…”

Solving the Inverse Problem of Electrocardiography on the Endocardium Using a Single Layer Source

“…The most prominent example of such statement is the equivalent double layer (EDL) defined on both epi- and endocardial surfaces of the heart (van Oosterom, 2014). According to the bidomain model (Tung, 1978), the EDL is proportional to the transmembrane potential when the body electrical conductivity as well as the extracellular and intracellular myocardial conductivities are considered to be isotropic and the sum of the extracellular and intracellular conductivities is equal to those of the body (Geselowitz, 1989; Kalinin et al, 2017). This electrophysiological meaning was shown to be highly beneficial for construction of ECGI-specific regularization techniques (Berger et al, 2006, 2011; van Dam et al, 2009).…”