Efficient identification of scars using heterogeneous model hierarchies.

AIMS: Detection and quantification of myocardial scars are helpful for diagnosis of heart diseases and for personalized simulation models. Scar tissue is generally characterized by a different conduction of excitation. We aim at estimating conductivity-related parameters from endocardial mapping data. Solving this inverse problem requires computationally expensive monodomain simulations on fine discretizations. We aim at accelerating the estimation by combining electrophysiology models of different complexity. METHODS AND RESULTS: Distributed parameter estimation is performed by minimizing the misfit between simulated and measured electrical activity on the endocardial surface, subject to the monodomain model and regularization. We formulate this optimization problem, including the modelling of scar tissue and different regularizations, and design an efficient solver. We consider grid hierarchies and monodomain-eikonal model hierarchies in a recursive multilevel trust-region method. With numerical examples, efficiency and estimation quality, depending on the data, are investigated. The multilevel solver is significantly faster than a comparable single level solver. Endocardial mapping data of realistic density appears to be sufficient to provide quantitatively reasonable estimates of location, size, and shape of scars close to the endocardial surface. CONCLUSION: In several situations, scar reconstruction based on eikonal and monodomain models differ significantly, suggesting the use of the more involved monodomain model for this purpose. Eikonal models can accelerate the computations considerably, enabling the use of complex electrophysiology models for estimating myocardial scars from endocardial mapping data.

Large scale simulation of pressure induced phase-field fracture propagation using Utopia.

Non-linear phase field models are increasingly used for the simulation of fracture propagation problems. The numerical simulation of fracture networks of realistic size requires the efficient parallel solution of large coupled non-linear systems. Although in principle efficient iterative multi-level methods for these types of problems are available, they are not widely used in practice due to the complexity of their parallel implementation. Here, we present Utopia, which is an open-source C++ library for parallel non-linear multilevel solution strategies. Utopia provides the advantages of high-level programming interfaces while at the same time a framework to access low-level data-structures without breaking code encapsulation. Complex numerical procedures can be expressed with few lines of code, and evaluated by different implementations, libraries, or computing hardware. In this paper, we investigate the parallel performance of our implementation of the recursive multilevel trust-region (RMTR) method based on the Utopia library. RMTR is a globally convergent multilevel solution strategy designed to solve non-convex constrained minimization problems. In particular, we solve pressure-induced phase-field fracture propagation in large and complex fracture networks. Solving such problems is deemed challenging even for a few fractures, however, here we are considering networks of realistic size with up to 1000 fractures.