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# Using Geodesics in Cell Cytometry

### Summary:

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

### Description:

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.

### Major Accomplishments:

Segmentation algorithms are compared. Often cell populations contain hundreds of cells and a dissimilarity matrix may have  ten thousand or more entries. Fast algorithms are needed and were developed to compute geodesic distances. Nonparametric statistical tests were compared.

Several articles  appear in IEEE Transactions on Medical Imaging and the CVPR 2015 Proceedings where more details can be found.

May 12,2015

itl

### Customers/Contributors/Collaborators:

John Elliott, Michael Halter
Biosystems and Biomaterials Division

Javier Bernal and Gunay Dogan
Applied and Computational Mathematics Division

### Staff:

Charles Hagwood, ITL

### Related Programs and Projects:

Computational Biology, Cell Systems Science
Contact

Charles Hagwood
301-975-2846
charles.hagwood@nist.gov
100 Bureau Drive, M/S 8980
Gaithersburg, MD 20899-8980