Extension of Integral Curves Estimation to a Time-Dependent Tensor Field Model

Numerous DT-MRI studies have been implemented along with the development of statistical and probabilistic methods and their applications in neuroscience due to the presence of background noise in diffusion measurements. However, similar researches from longitudinal DT-MRI data have leveraged existing longitudinal data analysis methods, not sufficiently addressed with its theoretical framework. We identify the problem of tracing repeatedly measured fiber trajectories in a longitudinal DT-MRI study and quantify its uncertainty in closed form. The idea behind this research is that if the repeatedly measured integral curves (i.e., fiber paths) at a certain location of the brain appear to follow the same pattern during the follow-up period then it is indicative of normal brain connectivity without damage and/or progressive deterioration in that region within the study period. We propose two estimators: (i) an estimator of the true integral curve using spatial and temporal information (ii) an estimator that measures the rate at which the integral curve changes with respect to the change of time during the follow-up period. The asymptotic behavior of these estimators is proven.