Persistent homology provides a multiscale representation of biomedical images by capturing higher-order topological features that reflect their underlying structural organization. However, the resulting topological summaries are typically used as predictors or features for classification and group comparisons rather than treated as primary variables of interest. We construct a generalized multilevel functional framework for analyzing persistent-homology summaries as longitudinal functional responses in repeated three-dimensional structural magnetic resonance imaging (MRI). Specifically, we represent topological features using Betti curves and model these curves as count-valued functional responses. A negative-binomial distribution accommodates the discrete and potentially overdispersed nature of Betti counts, while the multilevel formulation accounts for the dependence induced by repeated measurements and separates between-subject and within-subject sources of functional variation. Functional principal component analysis further evaluates the dominant modes of variation at each level. A Bayesian approach is used for joint estimation of the functional regression and multilevel functional principal components. We apply this modeling framework to longitudinal structural MRI data from the OASIS-2 study to investigate associations between brain topology and demographic and clinical characteristics, including age, gender, follow-up time, and dementia severity. The results demonstrate that the framework can capture covariate-associated variation across the filtration continuum while evaluating distinct sources and patterns of longitudinal variation across homology dimensions.

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