Predicting curvature evolution on biological surfaces from clinical imaging-derived area dilation: a closed-form...

Curvature evolution on a deforming surface is governed by the full change in the surface metric, but on biological surfaces captured by serial three-dimensional imaging, only the local area change is observable. The loss of the shear component leaves prediction of curvature evolution underdetermined from imaging alone. On the thoracic aorta, where curvature change marks disease progression, we derive a closed-form equation that predicts the change in integrated Gaussian curvature from the area dilation and initial geometry. The equation combines a conformal term in the area dilation with a leading anisotropy correction from the initial geometry. These two analytic levels, augmented by multi-scale spatial features at neighboring regions and a graph neural network trained on residuals, form a four-level nested predictor. On a synthetic aortic geometry under prescribed isotropic expansion, the equation recovers the analytic coefficient exactly. Across a continuum from pure expansion to pure shear, it holds $R^2\ge 0.71$ . On 236 paired thoracic aortic surfaces spanning dissection, aneurysm, traumatic injury, and non-pathologic controls, the equation recovers within-surface curvature change patterns with per-patient median Pearson $\tilde{r}=+0.495[+0.453,+0.540]$ and pooled $R^2=+0.238[+0.225,+0.250]$ , matching the graph neural network on the same inputs. The residual is a direct measurement of how far the observed growth field departs from conformality.