Optimal monodomain approximations of the bidomain equations Bjørn Fredrik Nielsen * , Tomas Syrstad Ruud, Glenn Terje Lines, Aslak Tveito Simula Research Laboratory, P.O. Box 134, 1325 Lysaker, Norway Abstract The bidomain equations are widely accepted to model the spatial distribution of the electrical potential in the heart. Although order optimal methods have been devised for discrete versions of these equations, it is still very CPU demanding to solve the equations numerically on a sufficiently fine mesh in 3D. Furthermore, the equations are hard to analyze; from a mathematical point of view, very little is known about the qualitative behavior of the solutions generated by these equations. It is well known that upon appealing to a certain relation between the extracellular and intracellular conductivities, the bidomain model can be rewritten in terms of a scalar reaction diffusion equation referred to as the monodomain model. This model is of course much easier to solve, and also the qualitative properties of the solutions are well known; such equa- tions have been studied intensively for decades. It is the purpose of the present paper to show how the bidomain equations can be approximated in an optimal manner by the solution of a monodomain model. The key feature here is that this optimal solution can be computed without solv- ing the bidomain model itself. The solution is obtained by putting the problem into a framework of parameter identifica- tion problems. Our results are illuminated by a series of numerical experiments. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Electrical behavior of cardiac tissue; Bidomain and monodomain models; Output least squares 1. Introduction The electrical activity in the heart is governed by a complex, fully coupled system of ordinary and partial differential equations: os ot ¼ F ðs; vÞ in H ; ð1Þ v t þ I ðv; sÞ¼rðM i rvÞþrðM i ruÞ in H ; ð2Þ rðM i rvÞ þ r  ððM i þ M e ÞruÞ¼ 0 in H : ð3Þ 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.158 * Corresponding author. E-mail addresses: bjornn@simula.no (B.F. Nielsen), tomassru@simula.no (T.S. Ruud), glennli@simula.no (G.T. Lines), aslak@ simula.no (A. Tveito). Applied Mathematics and Computation 184 (2007) 276–290 www.elsevier.com/locate/amc