Geometrical modeling of the heart - Rousseau, Olivier (University of Ottawa (Canada), 2010)
The heart is a very important human organ, that has a complex structure. Cardiovascular diseases have been the highest cause of death in North America and in Europe for decades. For this reason, a lot of research is made to understand the heart's physiology. One way to better understand the heart is via theoretical modeling of physiological mechanisms, the main ones being (1) trans-membrane potential wave propagation, (2) myocardium's contraction and (3) blood flow in the cardiac chambers. These physiological phenomena can be modeled via systems of partial differential equations (PDEs) that are defined on a domain given by the heart's shape. Numerical methods for solving these equations playa crucial role for validating these models. Numerical simulations also serve to make predictions of the organ's reaction to given stimuli. Thereby medical interventions such as the introduction of a pacemaker can be numerically simulated before attempting the surgical implantation. Such simulations on a complex geometry (the heart muscle or blood chambers) are usually made using the finite element or the finite volume methods. These methods requires a mesh of the computational domain, that is a triangulation of the domain into triangles in 2D or into tetrahedra in a 3D scenario. Thus far, most computations are made on meshes of idealized geometries and there is a lack of accurate 3D geometrical models of the heart. The community is aware of the importance of building accurate 3D models of the heart for understanding its physiology. There is only one realistic heart model that is publicly available. The main achievement of this project is to have built a precise and complete geometrical model of the human heart. The model consists of 1. An accurate and properly refined mesh of the heart muscle and chambers. The model includes fine features such as the pulpillary muscles (pillars) in the left and right ventricles. 2. The orientation of cardiac fibers. 3. The model is publicly available to the scientific community1. Other contributions of this thesis include a careful analysis, of known PDE-based segmentation methods. This is done in Chapter 5. We mostly studied the active contour without edges algorithm. We compared the impact of the choice of discretization on the numerical solutions of the problem. We concluded that some discretizations, while being more natural, do not behave as well as some others. We also evaluated the impact of the initial condition that is chosen on the speed of convergence of the algorithm. We carefully studied the hierarchical method of Gao and Bui [39], and show test cases where it performs better than the original multiphase algorithm of Vese and Chan [117]. We show that the hierarchical segmentation is a more natural framework for segmenting junctions of three segments. We also proposed modifications of some PDE-based methods to be able to do the heart segmentation. In Chapter 5, we introduced two new types of initial conditions that make the active contour without edges algorithm converge more quickly. Also the hierarchical segmentation algorithm with an L 1 fidelity term is introduced and is shown to be more efficient in some contexts. In Chapter 7, we present a variant of the subjective surface problem introduced by Sarti, Malladi and Sethian [98, 99]. We propose to solve the problem on an annulus around the heart chambers. During this project, we have developed C++ classes that handles 2D and 3D images. The result is a small PDE image processing toolkit (SPDEIPTK) that is suitable for research use. It features an almost transparent parallel implementation. The toolkit is also publicly available2. 1http://www.mathstat.uottawa.ca/∼orous272/ 2http://www.mathstat.uottawa.ca/∼orous272/spdeiptk/index.html
Keywords: Applied Mathematics.; Mathematics.