Cornelius, Nathan2014-02-252019-01-282014-01-27bibid: 8442331https://hdl.handle.net/1813/36142Mathematical models can serve as useful tools to better understand physiology and biological phenomena. This work outlines several mathematical models and their connection with various types of cortical microvascular topology and blood flow data obtained with various imaging modalities. Three models are proposed. The first is a model based on electrical circuit ideas that describes the relationship between cortical neural activity and space-resolved and time-resolved blood flows in the ensuing hemodynamic response. The second model, also based on electrical circuits, seeks to predict blood flows in a network of blood vessels based on topological network data and experimental blood flow measurements taken on a subset of vessels in the network. Finally, random graph ideas are used to propose two related models to represent the cortical microvasculature topology. The first is a Poisson process approach in which a vessel network is modeled by randomly positioning nodes in a three-dimensional space and randomly placing an edge between pairs of nodes based on various hard and soft constraints. The second related model is based on a Gibbsian Markov Random Field approach in which a vessel network is created using a Hamiltonian that favors or penalizes certain network features according to physiologic observations of vessel network topology. A wide range of applications of these types of models are demonstrated.en-USdissertation or thesis