Modélisation mathématique, simulation et réduction d’ordre de flux oculaires et leurs interactions : Construire le jumeau numérique de l’œil
Thomas Saigre · HAL (Le Centre pour la Communication Scientifique Directe) · 2024
The human body is a complex system, and the human eye is no exception. Despite significant advancements in medical research, many questions regarding ocular pathologies remain unanswered. The use of mathematical and computational models has revealed intricate mechanisms underlying human physiology. Due to its special connection with the brain, the eye is considered a window into the brain, providing non-invasive access to a range of biological markers that can aid in diagnosing neurodegenerative diseases. Therefore, understanding the eye's behavior, the diseases that affect it, and the potential treatments is crucial.This thesis focuses on the mathematical modeling and numerical simulation of ocular fluid dynamics within the human eye, particularly on heat transfer and aqueous humor flow. These methods must undergo validation with clinical data to ensure their reliability. Bio-physical models involve numerous parameters that may be patient-specific or influenced by external conditions. The objective of this sensitivity analysis is to understand how these parameters affect the eye's behavior and how they can be used for disease diagnosis, potentially assisting clinicians in selecting the best treatment. Such a study requires numerous simulations, which can be computationally expensive for the complex models used in this thesis. To reduce this computational cost, we have developed model reduction methods that decrease the number of equations to solve while maintaining result accuracy.In the first part of the thesis, we present the geometric and bio-physical models of the eyeball concerning heat transfer. Next, we discuss the numerical discretization methods implemented to simulate this model.The third chapter introduces the model reduction techniques used to lower the computational cost of these simulations, particularly through the certified reduced basis method. The fourth chapter addresses a specific issue related to the study of pointwise quantities of interest within the framework of reduced bases for elliptic problems with a Dirac source term. In the fifth chapter, we present the sensitivity analysis results obtained from our models. The sixth chapter extends the initial thermal model by incorporating the flow of aqueous humor in the anterior and posterior chambers of the eye. Finally, the seventh chapter provides an overview of the implementation contributions of this thesis.