Bioelectrical inverse problem

The first is the Dirichlet condition, which says that one has a set of discrete measurements of the voltage of a subset of the outer surface. The second is the natural Neumann condition. While it does not look much different from the formulation of the direct problem, the inverse formulations are ill-posed. The bioelectric inverse problem in terms of primary current sources does not have a unique solution, and the solution does not depend continuously on the data. Thus, to obtain useful solutions, one must try to... 

Y. Rudy and B.J. Messinger-Rapport. The inverse solution in electrocardiography solutions in terms of epicardial potentials. CRC Crit. Rev. Biomed. Eng., 16 215-268,1988. An excellent overview on the inverse problem in electrocardiography as well as a section on the application of the boundary element method to bioelectric field problems. 

C.R. Johnson and R.S. MacLeod. Nonuniform spatial mesh adaption using a posteriori error estimates applications to forward and inverse problems. Appl. Numer. Math., 14 331-326, 1994. This is a paper by the author which describes the apphcation of the h-method of mesh refinement for large scale two- and three-dimensional bioelectric field problems. 

In electrophysiology, a body surface potential is often recorded, with the purpose of characterizing the unknown bioelectric source in position, size, and direction. To go from measured potentials in a known conductive medium and calculate back to the source properties is called an inverse problem. Usually there are infinitely many possible solutions to a posed problem (see Figure 8.11(c)). Important classical patient examination methods such as electrocardiogram and electroencephalogram are aimed at characterizing properties of the source organ, and are thus based upon more or less empirical solutions to the inverse problem. 

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