Numerical methods for cardiovascular problems: computational electrocardiology and fluid dynamics in moving domains
Monday 11th January 2010
In this work we address the mathematical modeling and numerical simulation of cardiovascular phenomena, such as the propagation of the electrical signal in the heart tissue and the blood flow in large arteries. A strong motivation of this research activity comes from the interaction with the medical community that pinpoints the relevance and the complexity of medical issues occurring in the human circulatory system. The long term goal of this research is to set up a system of mathematical models, numerical techniques and software tools able to help the clinical practice by providing additional information so to support the medical decision process. Two important components to reach this goal are the integration between medical data and mathematical models and the promptness with whom patient-specific numerical results can be provided to medical doctors. This thesis should be regarded in this framework, focusing both on the development of accurate and efficient numerical methods for electrocardiology simulations, and on the integration of medical data on the motion of biological tissues within the numerical models and simulations. We first introduce the main features of the cardiovascular system, describing with more details the anatomy, the physiology and some pathological behaviors of the heart. In particular we present the organization of the myocardial tissue and the microscopical and macroscopical mechanisms that make the electrical signal propagate, since a large part of the present work concerns the effective numerical simulation of this phenomenon. We then focus on the mathematical models that are commonly used to describe the electrical signal propagation in the tissue. More precisely, models for electrocardiology result from the coupling between a cell level model, which reproduces the sudden variation of the potential across a single cell membrane, and a tissue level model, which describes how the transmembrane potential propagates in the tissue. The transmembrane potential variation in time (called action potential) is due to chemical processes at microscopical level and can be modeled by exploiting equations of different complexity, according to the level of detail to be reached. In the present work we focus on the simple Rogers-McCulloch model and the more complex Luo-Rudy phase I model. One of the most accurate models for the action potential propagation in the tissue is the Bidomain model, which describes both the extracellular and the intracellular domain, and the Monodomain model, a simplified version of the Bidomain one. Due to its mathematical structure, the Bidomain model is expensive to be solved numerically while the Monodomain model is less expensive but do not reproduce accurately the action potential propagation in some cases. Therefore two novel numerical methods have been developed during this work and are reported in the thesis, after a general presentation on the state of the art of numerical simulations for electrocardiology. In particular we first derive an ad-hoc preconditioner for the Bidomain, which is based on a proper reformulation of the Monodomain problem. This preconditioner is proven to be optimal with respect to the mesh size parameter and 3D numerical tests show that it leads to important reductions (even about 50%) of the CPU time required by the simulation, with respect to a standard strategy. The second technique we propose is a model adaptivity strategy, which consists in solving a hybrid model called Hybridomain, which corresponds to solve the Bidomain model in a partition of the computational domain and the Monodomain one in the remaining part. The choice of the partition is time dependent and is performed by means of a model error estimator which controls the difference between the solution of the Bidomain and the solution of Hybridomain model. The Bidomain is locally activated where the model error estimator is large and the Monodomain in solved elsewhere. 3D tests with LifeV library, both in healthy and unhealthy cases, show that this strategy, which can also be coupled to the use of the Monodomain preconditioner, allows to save computational time and to maintain a good accuracy of the results. In the last part of the present work we focus on developing and testing a pipeline to include the actual motion of biological structures in the simulation. This technique has a general validity, and it has been successfully applied in this work to the simulation of blood fluid dynamics in moving vessels. In particular from a set of medical images of a specific region acquired at different time frames during the cardiac cycle, we extract the domain of interest and we track the motion of each point in time, using a proper registration algorithm. To this extent, a registration algorithm called non-rigid viscous fluid registration is studied and implemented, obtaining promising preliminary results. The further step is the formulation of the problem equations in a moving domain framework. In the case of blood flow in arteries the natural choice is the Arbitrary Lagrangian Eulerian formulation (ALE). Finally the simulation of the interested phenomenon can be performed in a moving domain. This image-based motion strategy is applied to the modeling of blood flow in a patient-specific aortic arch geometry, comparing the results with a fixed-domain simulation. We also proposed a validation of the approach, through a comparison with the results of a more standard fluid-structure interaction algorithm. The technique is promising since the differences between the two computed solution fields are very small and the image-based motion numerical simulations require less computational effort than standard FSI strategies. The same pipeline can be applied to perform electrocardiology simulations in a moving heart, but the large displacements occurring in the heart make it necessary to devise ad hoc registration algorithms. Finally, since a large part of the present work has been devoted to develop and test numerical approaches, the ideation and implementation of the algorithms constitutes an important component of this work. For this reason we report in a separate Chapter of the thesis the most interesting aspects of the implementation of each method previously introduced.