An Eulerian PDE approach for computing tissue thickness

We outline an Eulerian framework for computing the thickness of tissues between two simply connected boundaries that does not require landmark points or parameterizations of either boundary. Thickness is defined as the length of correspondence trajectories, which run from one tissue boundary to the other, and which follow a smooth vector field constructed in the region between the boundaries. A pair of partial differential equations (PDEs) that are guided by this vector field are then solved over this region, and the sum of their solutions yields the thickness of the tissue region. Unlike other approaches, this approach does not require explicit construction of any correspondence trajectories. An efficient, stable, and computationally fast solution to these PDEs is found by careful selection of finite differences according to an up-winding condition. The behavior and performance of our method is demonstrated on two simulations and two magnetic resonance imaging data sets in two and three dimensions. These experiments reveal very good performance and show strong potential for application in tissue thickness visualization and quantification. Index Terms--Correspondence trajectory, numerical methods, partial differential equations (PDEs), thickness.