Longitudinal 3D Morphometry Study using the Optimal Mass Transport

In the longitudinal study of the progression of a disease or the recover process of a treatment plan, often we are interested in seeking the answers to three questions: From where, how the situation evolves to its current state; To where, how the situation is going to evolve in the future; and How far, the evolution of disease/recovery is not always on the same pace as time. Hence, given two or more instances along the evolutionary path, how far they are from each other in the morphometric sense. In this work, we model such morphometry evolution using the theory of optimally transportation of mass, and aim at providing the answers to the previous questions under such framework. Mathematically, we solve the optimal mass transport (OMT) problem by solving the Monge-Ampere equation. As a result, instead of few discrete sampling points separated by a fixed amount of time, the entire morphometric evolution path can be computed. In addition, we can further extrapolate the path to predict the future status of the disease or recovery. Thirdly, a measurement for "how advance is the disease" or "how well the tissue is recovering" can be defined naturally and rigorously. The proposed method is applied on the cases in the traumatic brain injury and Huntington disease and both qualitative and quantitative results are presented.