Optimal mass transport based brain morphometry for patients with congenital hand deformities

Congenital hand deformities (CHD) have attracted increasing research attention in the past few decades. The impacts of CHD on the brain structure, however, are not fully studied to date. In this work, we propose a novel framework to study brain morphometry in CHD patients using Wasserstein distance based on optimal mass transport (OMT) theory. We first employ conformal mapping to map the left and right surface-based functional brain areas to planar rectangles, which pushes the area element on the brain surface to the planar rectangle and incurs the area distortion. A measure is then determined by this area distortion. We further propose a new rectangle domain-based OMT map algorithm. Given two measures on two surfaces, we employ the proposed algorithm to compute a unique OMT map between the two measures encoding the geometric information of left and right surface-based functional brain areas. The transportation cost of this OMT map gives the Wasserstein distance between two surfaces, which intrinsically measures the dissimilarities between two surface-based shapes. Our method is theoretically rigorous and computationally efficient and stable. We finally evaluate the proposed Wasserstein distance-based method on the left and right post-central gyri from the CHD patients and healthy control subjects for analyzing brain cortical morphometry. Experimental results demonstrate the efficiency and efficacy of our method, and shed insightful lights on the study of the brain morphometry for those subjects with CHD.