Three-dimensional mapping of cortical thickness using Laplace's Equation

Abstract

We present a novel, computerized method of examining cerebral cortical thickness. The normal cortex varies in thickness from 2 to 4 mm, reflecting the morphology of neuronal sublayers. Cortical pathologies often manifest abnormal variations in thickness, with examples of Alzheimer's disease and cortical dysplasia as thin and thick cortex, respectively. Radiologically, images are 2-D slices through a highly convoluted 3-D object. Depending on the relative orientation of the slices with respect to the object, it is impossible to deduce abnormal cortical thickness without additional information from neighboring slices. We approach the problem by applying Laplace's Equation (V2psi = 0) from mathematical physics. The volume of the cortex is represented as the domain for the solution of the differential equation, with separate boundary conditions at the gray-white junction and the gray-CSF junction. Normalized gradients of psi form a vector field, representing tangent vectors along field lines connecting both boundaries. We define the cortical thickness at any point in the cortex to be the pathlength along such lines. Key advantages of this method are that it is fully three-dimensional, and the thickness is uniquely defined for any point in the cortex. We present graphical results that map cortical thickness everywhere in a normal brain. Results show global variations in cortical thickness consistent with known neuroanatomy. The application of this technique to visualization of cortical thickness in brains with known pathology has broad clinical implications.