Using Geodesics in Cell Cytometry

This  statistical application involves determining the similarity of cell populations based on comparing cell shapes.

This work is  being collaboratively done with John Elliott, Michael Halter both of the Biochemical Science Division, NIST and with Javier Bernal  of the Applied and Computational Mathematics Division.  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.  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.  Furthermore,  Srivastava et al have found numerical routines for computing geodesics. These numerical routines were developed into Mathematica code for this project. The development of the Mathematica code was done with the help of Tegan Brennan, a NIST summer SURF student from Princeton University. Below a geodesic path between two tells, 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, the cell shapes.  The segmentation was done using the Canny edge detector by Javier Bernal. Finally, the Mathematica ElasticGeodesic package is run to produce the geodesic and its length.

In our work, the Riemannian length of the geodesic path between cells are used to compare shapes. The Karcher mean shape of a group of shapes is computed.  The distances to the Karcher mean provide data for comparisons using  classical statistical methods.

Mathematica code to determine geodesics. Analysis of variance is used in conjunction with geodesic distances to test if shapes of cells grown on different substrates differ.