Cette thèse est consacrée à l’étude de quatre problèmes issus de la biologie. Le premier concerne la modélisation d’une population de métastases. Le modèle abouti a une équation de McKendrick-Von Foerster : une équation de conservation munie d’un terme au bord non–local. Nous montrons l’existence d’une unique solution et étudions son comportement asymptotique à l’aide de la notion d’entropie relative généralisée. L’étude numérique utilise le schéma WENO. Le deuxième concerne la modélisation de la respiration. Nous étudions la simulation des flux d’air dans l’appareil respiratoire à l’aide d’un modèle multi–échelle. Le système obtenu possède des conditions aux bords dissipatives non–usuelles. La méthode numérique employée est une méthode de décomposition qui permet de réduire le problème à la résolution de problèmes de Stokes avec conditions aux bords de type Dirichlet–Neumann classiques. Puis nous proposons un modèle pour les échanges gazeux montrant l’hétérogénéité de l’absorption de l’oxygène le long de l’arbre bronchique. La troisième partie concerne la cascade MAPK dans des ovocytes de Xénopes. La modélisation amène à une équation de type KPP. Après une étude mathématique montrant l’existence d’un front d’onde, nous réalisons une étude numérique fine du système. Enfin, nous étudions le système de Patlak–Keller–Segel 1D après explosion. Après une étude mathématique permettant de décrire le système après explosion à l’aide d’une mesure de défaut, nous donnons un schéma numérique adoptant le point de vue du transport optimal et permettant de simuler le système après explosion. / We investigate four models coming from biological contexts. The first one concerns a model describing the growth of a population of tumors. This model leads to a McKendrick–Von Foerster equation : a conservation law with a non–local boundary condition. We prove the existence and unicity of a solution, then we study, using the general relative entropy, its asymptotic behavior. We provide numerical simulations using WENO scheme. The second part concerns the modelisation of the respiration. First we study the air flux in the bronchial tree using a mulstiscale model. The system present non–usual dissipative boundary conditions. The numerical scheme we use is based on a decomposition idea that reduce the system to the resolution of Stokes problems with standard Dirichlet–Neumann conditions. Then, we propose a model concerning the gas exchanges bringing to light the heterogeneity of the absorption of oxygen along the bronchial tree. The third part concerns the MAPK cascade in Xenopus oocytes. The modelisation leads to an equation of KPP type. A mathematical study shows the existence of travelling waves. Then we provide a detailed numerical study of the system. Finally, the last part, concerns the system of Patlak–Keller–Segel 1D after blow–up. The mathematical study provide a description of the system after blow–up, based on the notion of default meausure. Then we propose a numerical scheme, adopting the optimal transport viewpoint and allowing to simulate the system after blow–up.

Characterized by excessive myocardial glycogen deposition, cardiac hypertrophy, frequent cardiac arrhythmias and progressive conduction system disease, the PRKAG2 cardiac syndrome stems from a genetic mutation in the γ2-subunit of AMP-activated protein kinase (AMPK). Although functionally diverse, the main role of AMPK is to modulate cardiac metabolism in response to depleted ATP levels. A comprehensive study of the dysfunctional regulation of AMPK activity involved in the progression of the human PRKAG2 cardiac syndrome is hindered by the limitations of in vitro techniques. Positron emission tomography (PET) imaging with the glucose analogue, FDG, offers a quantitative assessment of myocardial glucose uptake non-invasively. The aim of this thesis was to determine the ability of FDG to detect changes in glucose uptake, storage and metabolism in the heart in relation to AMPK activity and provide insights into the mechanism of PRKAG2 cardiac hypertrophy. To achieve this aim, a transgenic AMPK γ2-subunit Arg302Gln mouse model was evaluated with small animal FDG PET with correlation to biochemical assays of cardiac AMPK activity and the glycogen metabolism pathway. Using the vena cava blood input function, FDG myocardial glucose uptake was reliably assessed in mice for the first time with Patlak modeling. Reduced FDG uptake in the Arg302Gln PRKAG2 mouse model suggested a feedback pathway reducing exogenous glucose uptake due to excessive intracellular glycogen stores. Despite an increase in FDG uptake in the skeletal muscle of the PRKAG2 mutant mice following insulin stimulation, there was no change in cardiac uptake, signifying myocardial insulin resistance. Increased reliance on glucose oxidation by TMZ inhibition of fatty acid oxidation reduced glycogen stores, restored cardiac function and eliminated ventricular preexcitation. The observed reduction in mouse myocardial FDG uptake mirrors the reduction previously observed in the human PRKAG2 patients. The potential now exists to evaluate both progression and therapeutic interventions for the PRKAG2 cardiac syndrome with the transgenic mouse model with translation to the affected patients using FDG cardiac imaging.

Characterized by excessive myocardial glycogen deposition, cardiac hypertrophy, frequent cardiac arrhythmias and progressive conduction system disease, the PRKAG2 cardiac syndrome stems from a genetic mutation in the γ2-subunit of AMP-activated protein kinase (AMPK). Although functionally diverse, the main role of AMPK is to modulate cardiac metabolism in response to depleted ATP levels. A comprehensive study of the dysfunctional regulation of AMPK activity involved in the progression of the human PRKAG2 cardiac syndrome is hindered by the limitations of in vitro techniques. Positron emission tomography (PET) imaging with the glucose analogue, FDG, offers a quantitative assessment of myocardial glucose uptake non-invasively. The aim of this thesis was to determine the ability of FDG to detect changes in glucose uptake, storage and metabolism in the heart in relation to AMPK activity and provide insights into the mechanism of PRKAG2 cardiac hypertrophy. To achieve this aim, a transgenic AMPK γ2-subunit Arg302Gln mouse model was evaluated with small animal FDG PET with correlation to biochemical assays of cardiac AMPK activity and the glycogen metabolism pathway. Using the vena cava blood input function, FDG myocardial glucose uptake was reliably assessed in mice for the first time with Patlak modeling. Reduced FDG uptake in the Arg302Gln PRKAG2 mouse model suggested a feedback pathway reducing exogenous glucose uptake due to excessive intracellular glycogen stores. Despite an increase in FDG uptake in the skeletal muscle of the PRKAG2 mutant mice following insulin stimulation, there was no change in cardiac uptake, signifying myocardial insulin resistance. Increased reliance on glucose oxidation by TMZ inhibition of fatty acid oxidation reduced glycogen stores, restored cardiac function and eliminated ventricular preexcitation. The observed reduction in mouse myocardial FDG uptake mirrors the reduction previously observed in the human PRKAG2 patients. The potential now exists to evaluate both progression and therapeutic interventions for the PRKAG2 cardiac syndrome with the transgenic mouse model with translation to the affected patients using FDG cardiac imaging.