In this paper we study some mathematical properties of an inverse problem arising in connection with electrocardiograms (ECGs). More specifically, we analyze the possibility for recovering the transmembrane potential in the heart from ECG recordings, a challenge currently investigated by a growing number of groups. Our approach is based on the bidomain model for the electrical activity in the myocardium, and leads to a parameter identification problem for elliptic partial differential equations (PDEs). It turns out that this challenge can be split into two subproblems: (1) The task of recovering the potential at the heart surface from body surface recordings. (2) The problem of computing the potential inside the heart from the potential determined at the heart surface. Problem (1), which can be formulated as the Cauchy problem for an elliptic PDE, has been extensively studied and is well-known to be severely ill-posed. The main purpose of this paper is to prove that problem (2) is stable and well-posed if a suitable prior is available. Moreover, our theoretical findings are illuminated by a series of numerical experiments. Finally, we discuss some aspects of uniqueness related to the anisotropy in the heart.

@TechReport{BNM10,
  author       = {Burger, M. and Nielsen, B.-F. and Mardal, K.-A.},
  title        = {Stability analysis  of the inverse transmembrane potential problem in electrocardiography},
  institution  = {WWU Muenster},
  year         = {2010},
  url          = \{/2010/BNM10},
}