An a posteriori error estimator for model adaptivity in electrocardiology L. Mirabella a,b,⇑ , F. Nobile a , A. Veneziani b a MOX (Modeling and Scientific Computing), Department of Mathematics, Politecnico di Milano, P.za L. da Vinci 32, 20133 Milano, Italy b Department of Mathematics and Computer Science, Emory University, 400 Dowman Dr NE, 30322 Atlanta, GA, USA article info Article history: Received 10 August 2009 Received in revised form 15 January 2010 Accepted 8 March 2010 Available online 23 March 2010 2000 MSC: 65M60 92C05 Keywords: Computational electrocardiology Model adaptivity A posteriori error estimation Modeling error abstract We introduce an a posteriori modeling error estimator for the effective computation of electric potential propagation in the heart. Starting from the Bidomain problem and an extended formulation of the sim- plified Monodomain system, we build a hybrid model, called Hybridomain, which is dynamically adapted to be either Bi- or Monodomain ones in different regions of the computational domain according to the error estimator. We show that accurate results can be obtained with the adaptive Hybridomain model with a reduced computational cost compared to the full Bidomain model. We discuss the effectivity of the estimator and the reliability of the results on simulations performed on real human left ventricle geometries retrieved from healthy subjects. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction The propagation of electrical potential in the cardiac tissue is well described by the so-called Bidomain model (see e.g. [22]), which is a system of non-linear unsteady partial differential equations coping with both the intra- and extracellular potential dynamics. Usually, the computational cost of numerical simulations of this system is high due to the degenerate parabolic nature of the model, being the time derivative vector multiplied by a singular matrix. Moreover, an accurate solution on real geometries demands for fine meshes and time steps. For these reasons, many applications con- sider a simplified model called Monodomain. It relies however on an assumption on the fibers conductivity which is not always veri- fied and this model is not able to predict certain physiological and pathological patterns, especially in the neighborhood of a propagat- ing front [13]. Moreover, the standard Monodomain model does not predict correctly the front propagation velocity [20]. See also [7,8] for an ‘‘improved” Monodomain model that features a front speed closer to the Bidomain one. In this work we will refer to the standard Monodomain model as described e.g. in [7]. Recent literature has been devoted to the efficient solution of the discretized Bidomain model and, in particular, to the develop- ment of efficient preconditioners (see e.g. [7,18,26,27,30,31]). In [12] an extended version of the Monodomain model has been pro- posed as a preconditioner for solving the Bidomain system. In this paper we follow a different approach for simulating po- tential propagation in the heart. More precisely, inspired by the re- cent literature on modeling error estimation and adaptation (see e.g. [3,17,19,21,28]), we combine the Bi- and Monodomain models in a model adaptivity framework. The basic idea is to confine the (more expensive) Bidomain solution to a small part of the domain at hand, while on its most part we solve the Monodomain equa- tion. In this way, we reduce the computational time, without sig- nificantly affecting the reliability of the numerical solution. The crucial step in this approach is the setup of a modeling error estimator able to identify the region where it is worth solving the Bidomain system. Based on the error estimate we solve a finite ele- ment discretization of the hybrid model. We actually solve the Bidomain model on some elements while in the most part of the domain we keep on solving the Monodomain system. Numerical results presented here are carried out on a real geometry retrieved from medical images and show that the hybrid model driven by our estimator is able to capture the most important features of the potential propagation described by a full Bidomain model with a good effectivity and CPU time reduction. The paper is organized as follows. In Section 2 we introduce the Bidomain and Monodomain systems and recall their features. We introduce the extended formulation of the Monodomain model and the Hybridomain system used for the model adaptivity. Moreover, we present the semi-discretization of these problems 0045-7825/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2010.03.009 ⇑ Corresponding author at: Department of Mathematics and Computer Science, Emory University, 400 Dowman Dr NE, 30322 Atlanta, GA, USA. E-mail address: lucia@mathcs.emory.edu (L. Mirabella). Comput. Methods Appl. Mech. Engrg. 200 (2011) 2727–2737 Contents lists available at ScienceDirect Comput. Methods Appl. Mech. Engrg. journal homepage: www.elsevier.com/locate/cma