Multi-manifold diffeomorphic metric mapping for aligning cortical hemispheric surfaces.

Cortical surface-based analysis has been widely used in anatomical and functional studies because it is geometrically appropriate for the cortex. One of the main challenges in the cortical surface-based analysis is to optimize the alignment of the cortical hemispheric surfaces across individuals. In this paper, we introduce a multi-manifold large deformation diffeomorphic metric mapping (MM-LDDMM) algorithm that allows simultaneously carrying the cortical hemispheric surface and its sulcal curves from one to the other through a flow of diffeomorphisms. We present an algorithm based on recent derivation of a law of momentum conservation for the geodesics of diffeomorphic flow. Once a template is fixed, the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point on curves and surfaces along the geodesic is completely determined by the momentum at the origin. We solve for trajectories (geodesics) of the kinetic energy by computing its variation with respect to the initial momentum and by applying a gradient descent scheme. The MM-LDDMM algorithm optimizes the initial momenta encoding the anatomical variation of each individual relative to a common coordinate system in a linear space, which provides a natural scheme for shape deformation average and template (or atlas) generation. We applied the MM-LDDMM algorithm for constructing the templates for the cortical surface and 14 sulcal curves of each hemisphere using a group of 40 subjects. The estimated template shape reflects regions which are highly variable across these subjects. Compared with existing single-manifold LDDMM algorithms, such as the LDDMM-curve mapping and the LDDMM-surface mapping, the MM-LDDMM mapping provides better results in terms of surface to surface distances in five predefined regions.