PDE-based reconstruction of the cerebral cortex from MR images

The topologically correct and geometrically accurate reconstruction of the cerebral cortex from magnetic resonance (MR) images is an important step in quantitative analysis of the human brain structure, e.g. in cortical thickness measurement studies. Limited resolution of MR images, noise, intensity inhomogeneities, and partial volume effects can all contribute to geometrical inaccuracies and topological errors in the model of cortical surfaces. For example, unresolved touching banks of gray matter (GM) in narrow sulci pose a particular challenge for an automated algorithm, requiring specific steps for the recovery of separating boundaries. We present a method for the automated reconstruction of the cortical compartment from MR images. The method is based on several partial differential equation (PDE) modelling stages. First, a potential field is computed in an electrostatic model with GM posing as an insulating dielectric layer surrounding a charged conductive white matter (WM) object. Second, geodesic distances from WM along the streamlines of the potential field are computed in a Eulerian framework PDE. Third, a digital skeleton surface separating GM sulcal banks is derived by finding shocks in the distance field. At the last stage, a geometric deformable model based on the level set PDE is used to reconstruct the outer cortical surface by advection along the gradient of the distance or potential field. The rule preserving the digital topology, and the skeleton of the distance field resolving fused adjacent banks in sulci, constrain the deformable model evolution. In addition, the deformable model may use the distance field as a constraint on thickness of the reconstructed cortical layer.