Using Geodesics in Cell Cytometry

This project involves image cytometry, mathematics and statistics.  Understanding cellular responses to stimuli, such as pharmaceuticals or environmental toxins and discovering correlations between cellular responses and the cell phenotype are applications motivating this work. Change in shape is a known response of a cell to a stimuli.  Also, using image analysis and visualization software for tracking live cells developed at by the Cell Systems Science Group, we have begun using our results to track, over time, the relationship between cellular characteristics (expressed by GFP intensity) and cell shapes changes.

 Because cell shape is an infinite dimensional descriptor, the mathematics involved is non-routine. In particular, when correctly formalized, shape space is a differential manifold having a Riemannian metric. On this differential manifold, a line between to points in usual Euclidean space is generalized to a geodesic path between two shapes in shape space. The Riemannian length of this geodesic path provides a measure of the distance between two shapes. These methods have been formalized by recent work of Srivastava et al (2004). 

 Below a geodesic path between two cells, an NIH 3T3 mouse fibroblasts cell and an A10 rat vascular smooth muscle cell is shown. The geodesic length of this path is 0.75.

Several steps are involved in going from cells to geodesics. The process starts with fluorescence microscope images of cell populations. These images are then segmented to produce cell boundaries. Then, a numerical algorithm is used to determine the geodesic distance between the shapes of  two boundary contours.

For two populations of cells, dissimilarity matrices are created whose entries are geodesic distances between cell shape pairs. We used three nonparametric tests, the Friedmann-Rafsky Minimum Spanning Tree test, the Schilling Nearest Neighbor test and an energy test to test the hypothesis that two cell populations are statistically equivalent.