Unified Cortical Surface Morphometry and Its Application to Quantifying Amount of Gray Matter

In quantifying the amount of gray matter in a population, voxelbased morphometry (VBM) and cortical thickness analysis are the two most widely used techniques. There are still many unanswered methodological questions regarding these methods. It is not clear what is the optimal amount of registration needed in VBM. It is assumed that the gray matter density obtained in the VBM will be positively correlated to cortical thickness but there is no systematic study that compared these two disparate measurements. The gray matter density is a 3D measurement defined in voxels in 3D whole brain volume while the thickenss is a 1D measurement defined along the 2D cortical surface. Due to this disparate dimensionality, it is not clear how to compare them in a systematic fashion. In this paper, we present a novel weighted Fourier Series (WFS) representation for the cortical surface that enables us to address these questions in a unified mathematical framework. The WFS representation is a data smoothing technique formulated as a solution of the Cauchy problem. This representation provides the explicit smooth functional estimation of true unknown cortical boundary as a linear combination of basis functions. Based on this new representation, cortical thickness and gray matter density can be defined and compared naturally. The basic theory underlying the weighted Fourier Series representation and its numerical implementation issues are presented in detail. As an illustration, this unified approach is applied in the cortical thickness analysis and the VBM in a group of autistic subjects. The thickness analysis and VBM results are compared in parallel. 2