  Robust forecasting of the future anatomical changes inflicted by an ongoing
disease is an extremely challenging task that is out of grasp even for
experienced healthcare professionals. Such a capability, however, is of great
importance since it can improve patient management by providing information on
the speed of disease progression already at the admission stage, or it can
enrich the clinical trials with fast progressors and avoid the need for control
arms by the means of digital twins. In this work, we develop a deep learning
method that models the evolution of age-related disease by processing a single
medical scan and providing a segmentation of the target anatomy at a requested
future point in time. Our method represents a time-invariant physical process
and solves a large-scale problem of modeling temporal pixel-level changes
utilizing NeuralODEs. In addition, we demonstrate the approaches to incorporate
the prior domain-specific constraints into our method and define temporal Dice
loss for learning temporal objectives. To evaluate the applicability of our
approach across different age-related diseases and imaging modalities, we
developed and tested the proposed method on the datasets with 967 retinal OCT
volumes of 100 patients with Geographic Atrophy, and 2823 brain MRI volumes of
633 patients with Alzheimer's Disease. For Geographic Atrophy, the proposed
method outperformed the related baseline models in the atrophy growth
prediction. For Alzheimer's Disease, the proposed method demonstrated
remarkable performance in predicting the brain ventricle changes induced by the
disease, achieving the state-of-the-art result on TADPOLE challenge.
