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Monitoring treatment response in longitudinal studies plays an important role in clinical practice. Accurately identifying lesions across serial imaging follow-up is the core to the monitoring procedure. Typically this incorporates both image and anatomical considerations. However, matching lesions manually is labor-intensive and time-consuming. In this work, we present deep lesion tracker (DLT), a deep learning approach that uses both appearance- and anatomical-based signals. To incorporate anatomical constraints, we propose an anatomical signal encoder, which prevents lesions being matched with visually similar but spurious regions. In addition, we present a new formulation for Siamese networks that avoids the heavy computational loads of 3D cross-correlation. To present our network with greater varieties of images, we also propose a self-supervised learning (SSL) strategy to train trackers with unpaired images, overcoming barriers to data collection. To train and evaluate our tracker, we introduce and release the first lesion tracking benchmark, consisting of 3891 lesion pairs from the public DeepLesion database. The proposed method, DLT, locates lesion centers with a mean error distance of 7 mm. This is 5% better than a leading registration algorithm while running 14 times faster on whole CT volumes. We demonstrate even greater improvements over detector or similarity-learning alternatives. DLT also generalizes well on an external clinical test set of 100 longitudinal studies, achieving 88% accuracy. Finally, we plug DLT into an automatic tumor monitoring workflow where it leads to an accuracy of 85% in assessing lesion treatment responses, which is only 0.46% lower than the accuracy of manual inputs.