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Longitudinal MRIs are often used to capture the gradual deterioration of brain structure and function caused by aging or neurological diseases. Analyzing this data via machine learning generally requires a large number of ground-truth labels, which are often missing or expensive to obtain. Reducing the need for labels, we propose a self-supervised strategy for representation learning named Longitudinal Neighborhood Embedding (LNE). Motivated by concepts in contrastive learning, LNE explicitly models the similarity between trajectory vectors across different subjects. We do so by building a graph in each training iteration defining neighborhoods in the latent space so that the progression direction of a subject follows the direction of its neighbors. This results in a smooth trajectory field that captures the global morphological change of the brain while maintaining the local continuity. We apply LNE to longitudinal T1w MRIs of two neuroimaging studies: a dataset composed of 274 healthy subjects, and Alzheimer's Disease Neuroimaging Initiative (ADNI, $N=632$). The visualization of the smooth trajectory vector field and superior performance on downstream tasks demonstrate the strength of the proposed method over existing self-supervised methods in extracting information associated with normal aging and in revealing the impact of neurodegenerative disorders. The code is available at \url{https://github.com/ouyangjiahong/longitudinal-neighbourhood-embedding.git}.

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Longitudinal MRIs are often used to capture the gradual deterioration of brain structure and function caused by aging or neurological diseases. Analyzing this data via machine learning generally requires a large number of ground-truth labels, which are often missing or expensive to obtain. Reducing the need for labels, we propose a self-supervised strategy for representation learning named Longitudinal Neighborhood Embedding (LNE). Motivated by concepts in contrastive learning, LNE explicitly models the similarity between trajectory vectors across different subjects. We do so by building a graph in each training iteration defining neighborhoods in the latent space so that the progression direction of a subject follows the direction of its neighbors. This results in a smooth trajectory field that captures the global morphological change of the brain while maintaining the local continuity. We apply LNE to longitudinal T1w MRIs of two neuroimaging studies: a dataset composed of 274 healthy subjects, and Alzheimer's Disease Neuroimaging Initiative (ADNI, $N=632$). The visualization of the smooth trajectory vector field and superior performance on downstream tasks demonstrate the strength of the proposed method over existing self-supervised methods in extracting information associated with normal aging and in revealing the impact of neurodegenerative disorders. The code is available at \url{https://github.com/ouyangjiahong/longitudinal-neighbourhood-embedding.git}.

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