Laplace-based cortical thickness without volumetric discretisation

Abstract

Laplace-based methods define cortical thickness as the length of a streamline connecting the pial and white surfaces, but existing implementations discretize the volume and lose accuracy in tight folds. Surface-based methods use Euclidean inter-surface distances that cannot enforce orthogonal, non-crossing correspondences. We solve Laplace’s equation on the cortical surface itself using a boundary element method, with pial and white surfaces as Dirichlet boundaries and the surrounding edge Neumann boundaries; thickness is the length of a streamline traced along the analytic gradient field. We tested 8044 streamlines across analytical shapes, synthetic gyral patches, and real cortices; reciprocal paths returned within 0.05 mm of their origin, with 95th-percentile asymmetry below 0.68%. On folded patches with known ground truth, length-normalized bias was 0.0003 mm/mm independent of curvature. Streamline validity remained near-complete above a mean triangle shape quality of 0.90 and degraded sharply below 0.80, defining a practical mesh threshold. In three focal cortical dysplasia patients, each imaged at 3 T and 7 T, the method resolved focal thickening at the lesion site, with contralateral cortex mapped alongside as a visual reference. Vertex-wise comparison against FreeSurfer on identical surfaces showed differences scaling with local curvature. The observed bias is over an order of magnitude below preclinical Alzheimer’s thinning rates.