This dissertation focuses on the theoretical and applied aspects of a spatio-temporal modeling for the reconstruction of in-vivo fiber tracts in white matter when a single brain is scanned with magnetic resonance imaging (MRI) on several occasions. The objective of this research is twofold: one is how to estimate the spatial trajectory of a nerve fiber bundle at a given time point in the presence of measurement noise and the other is how to incorporate a progressive deterioration of brain connectivity into a hypothesis test. This dissertation leverages the spatio-temporal behavior of water diffusion in a region of the brain where the estimation of fiber trajectories is made from smoothing the time-varying diffusion tensor field via the Nadaraya-Watson type kernel regression estimator to its eigenvector field. The estimated fiber pathway takes the form of confidence ellipsoids given the estimates of mean and covariance functions. Furthermore, this dissertation proposes a hypothesis test in which the null hypothesis states that true fiber trajectories remain the same over a certain time interval. This null hypothesis indicates no substantial pathological changes of fiber pathways in that region of the brain during the observed time period. The proposed test statistic is shown to follow the limiting chi-square distribution under the null hypothesis. The power of the test is illustrated via Monte Carlo simulations. Lastly, this dissertation demonstrates the test can also be applied to a real longitudinal DTI study of a single brain repeatedly measured across time.
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