Thèse de doctorat--Sciences naturelles--Faculté des sciences de l'Université de Paris, 1901. N°: 7.

Reproduction de : Thèse d'exercice : Médecine. Pneumologie : Nantes : 2008. / Bibliogr.

Ce travail est consacré à la recherche de surfaces de Riemann (\it compactes) extrê\-mes (i.e. maxima locaux) pour la systole, ou tout au moins parfaites. En genre 4, on donne une nouvelle surface extrême et deux surfaces parfaites non extrêmes (ce sont les premiers exemples de telles surfaces en genre $\leq 10$). La méthode consiste à réaliser géométriquement les groupes d'automorphismes à 4 points de branchements. En effet, le lieu des points fixes dans l'espace de Teichmüller $T_g$ d'un tel groupe, dépend d'un paramètre complexe qu'on peut alors ajuster pour maximiser la systole. On étudie ensuite les propriétés variationnelles dans $T_g$ des surfaces obtenues. Par extension de cette méthode, on trouve également une nouvelle surface extrême en genre 6, ainsi qu'une suite infinie de surfaces parfaites non extrêmes de genre $g>3$. En outre, on retrouve, de manière unifiée, les surfaces déjà connues en genre $\leq 5$. La méthode employée pour la recherche de surfaces parfaites, permet de trouver parallèlement un certain nombre de surfaces eutactiques, qui sont intéressantes à classifier en elles-mêmes puisque ce sont les points critiques de la fonction systole. Enfin, le dernier chapitre, développant une toute autre approche, concerne une méthode purement algébrique qui permet de redémontrer l'extrémalité des surfaces respectivement de Bolza et de Klein.

Reproduction de : Thèse d'exercice : Médecine : Nancy 1 : 2005. / Titre provenant de l'écran-titre.

Diss. (sammanfattning) Stockholm : Karol. inst., 2006. / Härtill 5 uppsatser.

Understanding the mechanisms governing left ventricular function and dysfunction is critical to analyze cardiovascular disorders and gaining insights into possible therapies. Left ventricular function can be evaluated using Magnetic Resonance Imaging (MRI). Cardiac displacements and corresponding strains are then computed from the imaging data. In measuring and assessing the left ventricle’s motion, images are taken either in the short axis (top-down) or long axis (side) views. In this project, we will use DENSE MRI data, which measures the displacements of individual tissue voxels during the cardiac cycle. After extracting the myocardial tissue by segmenting the MR images, strains are computed by differentiating the displacement field in the radial direction (across the thickness of the heart wall), longitudinal direction (along the left ventricle long axis), and in the circumferential direction. Current approaches focus mainly on evaluating cardiac motion and strains during ventricular systole, when the ventricles contract and blood is pumped out of the heart ~\cite{srichai2009cardiovascular}. Our aim is to characterize strains during atrial systole, which corresponds to the late filling of the ventricles before the next contraction occurs. Understanding the deformation of the left ventricle during late filling is particularly important to evaluate the passive response of the myocardium, which is related to several cardiac diseases, such as heart failure with preserved ejection fraction and diabetic cardiomyopathy. During this study we will use preclinical data already acquired in healthy swine subjects. Our goal is to evaluate inter subject variability at peak atrial systole and how different segmentations (intra and inter observer variability) affect the computed strains.

Soit G un groupe de présentation finie. Un résultat de Gromov affirme l'existence de cycles géométriques réguliers qui représentent une classe d'homologie non nulle h dans le énième groupe d'homologie à coefficients entiers de G, cycles géométriques dont le volume systolique est aussi proche que souhaité du volume systolique de h. Ce théorème, dont aucune démonstration exhaustive n'avait été faite, a servi à obtenir plusieurs résultats importants en géométrie systolique. La première partie de cette thèse est consacrée à une démonstration complète de ce résultat. L'utilisation de ces cycles géométriques réguliers est connue sous le nom de technique de régularisation. Cette technique permet notamment de relier le volume systolique de certaines variétés fermées à d'autres invariants topologiques de ces variétés, tels que les nombres de Betti ou l'entropie minimale. La seconde partie de cette thèse propose d'examiner ces relations, et la mise en oeuvre de la technique de régularisation.La troisième partie est consacrée à trois problèmes liés à la géométrie systolique. Dans un premier temps on s'intéresse à une inégalité concernant les tores pleins plongés dans l'espace tridimensionnel. Puis, on s'intéresse ensuite aux triangulations minimales des surfaces compactes, afin d'obtenir des informations sur le volume systolique de ces surfaces. Enfin, on présente la notion de complexité simpliciale d'un groupe de présentation finie, et ses liens avec la géométrie systolique. / Let G be a finitely presented group. A theorem of Gromov asserts the existence of regular geometric cycles which represent a non null homology class h in the nth homology group with integral coefficients of G, geometric cycles which have a systolic volume as close as desired to the systolic volume of h. This theorem, of which no complete proof has been given, has lead to major results in systolic geometry. The first part of this thesis is devoted to a complete proof of this result.The regularizationtechnique consists in the use of these regular geometric cycles to obtain information about the class $h$. This technique allows to link the systolic volume of some closed manifolds to homotopical invariants of these manifolds, such as the minimal entropy and the Betti numbers. The second part of this thesis proposes to investigate these links.The third part of this thesis is devoted to three problems of systolic geometry. First we are investigating an inequality about embeded tori in $R^3$. Second, we are looking into minimal triangulations of compact surfaces and some information they can provide in systolic geometry. And finally, we are presenting the notion of simplicial complexity of a finitely-presented group and its links with the systolic geometry.

Dans le cadre de la géométrie riemannienne globale sans hypothèse de courbure en lien avec la topologie, nous nous intéressons au volume maximal des boules de rayon fixé dans les revêtements universels des graphes et des surfaces. Dans la première partie, nous prouvons que si l’aire d’une surface riemannienne fermée M de genre g ≥ 2 est suffisamment petite par rapport à son aire hyperbolique, alors pour chaque rayon R ≥ 0, le revêtement universel de M contient une R-boule d’aire au moins l’aire d’une cR-boule dans le plan hyperbolique, où c ∈ (0; 1) est une constante universelle. En particulier (quitte à prendre l’aire de la surface encore plus petite), nous démontrons que pour chaque rayon R ≥ 1, le revêtement universel de M contient une R-boule d’aire au moins l’aire d’une R-boule dans le plan hyperbolique. Ce résultat répond positivement pour les surfaces, à une question de L. Guth. Nous démontrons également que si Γ est un graphe connexe de premier nombre de Betti b ≥ 2 et de longueur suffisamment petite par rapport à la longueur d’un graphe trivalent Γb de premier nombre de Betti b dont la longueur de chaque arête est 1, alors pour chaque rayon R ≥ 0, le revêtement universel de Γ contient une R-boule d’aire au moins c fois l’aire d’une R-boule dans le revêtement universel de Γb, où c ∈ ( ½ ; 1). / This thesis deals with global Riemannian geometry without curvature assumptions and its link to topology, we focus on the maximal volume of balls of fixed radius in the universal covers of graphs and surfaces. In the first part, we prove that if the area of a closed Riemannian surface M of genus at least two is sufficiently small with respect to its hyperbolic area, then for every radius R ≥ 0 the universal cover of M contains an R-ball with area at least the area of a cR-ball in the hyperbolic plane, where c ∈ (0; 1) is a universal positive constant. In particular (taking the area of M smaller if needed), we prove that for every radius R ≥ 1, the universal cover of M contains an R-ball with area at least the area of a ball with the same radius in the hyperbolic plane. This result answers positively a question of L. Guth for surfaces. We also prove an analog result for graphs. Specifically, we prove that if Γ is a connected metric graph of first Betti number b ≥ 2 and of length sufficiently small with respect to the length of a connected trivalent graph Γb of the same Betti number where the length of each edge is 1, then for every radius R ≥ 0 the universal cover of Γ contains an R-ball with length at least c times the length of an R-ball in the universal cover of Γb, where c ∈ ( ½ ; 1) is a universal constant.

Thesis (Ph. D.)--University of Florida, 2003. / Title from title page of source document. Includes vita. Includes bibliographical references.

Heart failure is the clinical syndrome accompanying the inability of the heart to maintain a cardiac output required to meet the metabolic requirements and accommodate venous return, and is one of the leading causes of mortality in United States. Accurate imaging of the heart and its failure is important for successful patient management and treatment. Multiple cardiac imaging modalities provide complementary information about the heart – LV function and wall motion, anatomy, myocardial viability and ischemia. In many instances, it is necessary for a patient to undergo multiple imaging sessions to obtain diagnostic clinical information with confidence. It would be beneficial to the individual and the health care system if a single imaging modality could yield reliable clinical information about the heart, leading to a reduced cost, anxiety and an increased diagnostic confidence. This thesis proposes methods that would make cardiac MRI perform an improved assessment of LV function, wall motion, and viability, such that cardiac MRI is taken one step closer to being a single stop solution for imaging of heart. Conventional cardiac MR imaging is performed at a temporal resolution of around 40 ms per cardiac phase. While the global left ventricular (LV) function can be reliably established at this temporal resolution, functional metrics characterizing transient function like peak filling and ejection rates are not accurately assessed. A high temporal resolution is necessary to characterize such transient LV function and wall motion mechanics. This thesis proposes methods to acquire cine-images of the heart at a higher temporal resolution (~ 6 ms) and algorithms to acquire the LV volume across all cardiac phases that would yield functional metrics characterizing LV function and wall motion mechanics. The validation of these algorithms was performed on human subjects. Cardiac MR imaging is the current gold standard of myocardial viability imaging, in which scarred regions of the heart following myocardial infarction are visualized. However viability imaging faces image quality challenges in patients with severe arrhythmias and in cases where a higher spatial resolution, and hence a longer acquisition time, is desired. This thesis also proposes an arrhythmia insensitive inversion recovery (AIIR) algorithm that would significantly reduce artifacts that degrade image quality, thereby extending viability imaging to higher spatial resolution and in patients with severe arrhythmia. Simulations, experimental validation on phantoms and clinical verification on patients are performed. Results from high temporal resolution imaging reveal that obtaining cine cardiac MR images at a temporal resolution of 6 ms per cardiac phase is feasible. Appropriate validated algorithms yield LV time-volume curve from which LV functional metrics are reliably extracted. A dependence on temporal resolution is revealed, and a temporal resolution cut-off of 12 ms is proposed to reliably capture the temporal dynamics of the LV. Also, results from cardiac viability imaging show that the AIIR algorithm performs significantly better than conventional imaging methods in both phantoms and human subjects, as shown by the blinded expert scores, leading to a better image quality. In conclusion, this thesis proposes and implements methods that help cardiac MRI yield 1) a better function and wall motion assessment of the heart through high temporal resolution imaging and 2) a better assessment of myocardial viability through the AIIR algorithm.