Predicting Brain Morphometry with MT-GNN: Mesh Evolution in Continuous Time with Graph-Based Metric Tensor Embeddings

Predicting how a subcortical structure's shape will evolve from a few prior scans could support prognosis and clinical-trial enrichment. Existing longitudinal mesh predictors either extrapolate shape trajectories via high-dimensional embeddings or regress vertex deformations directly. We instead predict the surface's intrinsic geometry in continuous time: a single per-structure graph network predicts the future per-vertex first fundamental form (metric tensor) for an arbitrary causal multiple-visit history and an arbitrary prediction horizon, conditioned on a Fourier encoding of the lead time. The predicted metric is decoded into a surface by a differentiable As-Rigid-As-Possible solver, and the model is trained end-to-end on the rigid-aligned vertex error. Training through the reconstruction keeps the decoded prediction a valid surface and consistently improves it. On 14 subcortical structures from the ADNI dataset, the proposed mesh evolution model (MT-GNN) predicts best among the evaluated methods at every horizon (-2.29% mean vertex error vs. the temporal mean, p=6.1×10-5, beating it on 14/14 structures), ahead of geodesic shape regression (DCM, -0.19%) and a mesh transformer (TransforMesh, -0.45%; p=1.2×10-4), with the lead widening as the horizon grows.