PROBABILITY STATE MODELING: A NEW PARADIGM FOR CYTOMETRIC ANALYSIS - Flow Cytometry in Drug Discovery and Development - page 291

14.12 BACK TO PARAMETRIC EQUATIONS

We began this chapter with a set of parametric equations that defined a line in
n-dimensional space. Most of us initially learned that a line had the following simple
equation:

y ¼ mx þ b

A slope, m, multiplies the independent variable, x, and the intercept, b, is added to
the product to form the dependent variable, y. A line represents a linear relationship
between x and y. What we learned was valuable and we have all used this relation at
some time.
What happens to this simple equation if we are interested in a line defined in three-
dimensional space? If you do not have a background in mathematics, you might be
tempted to think about the solution in terms of the three projections to two-
dimensional space.

y ¼ m 1 x þ b 1

z ¼ m 2 x þ b 2

z ¼ m 3 y þ b 3

But it should be clear that as we increase the number of dimensions, the number of
2d projections will increase geometrically. What seemed simple and intuitive in two-
dimensional space becomes a nightmare when we dramatically increase the number
of dimensions. The parallel with cytometry s history is striking. Parametric equations

have been used for over 200 years for a variety of applications, including reducing the
complexity of high dimensional mathematical constructs.

I think it is time for cytometry to make the same leap.

REFERENCES

1. Protter MH, Morrey CB. College Calculus with Analytic Geometry, Loomis LH, editor,
Addison-Wesley Series in Mathematics, Addison-Wesley, 1967, pp. 379-384.
2. Loken MR, Wells DA. Normal antigen expression in hematopoiesis. In: Stewart CC,
Nicholson JK, editors, Immunophenotyping, Wiley-Liss, 2000, pp. 38-142.
3. Loken MR, Shah VO, Danttilio KL, Civin CI. Flow cytometric analysis of human bone
marrow. II. Normal B lymphocyte development. Blood 1987;70:1316-1324.
4. Sanchez L, ThieffyD. A logical analysis of the Drosophila gap-gene system. J. Theor. Biol.
2001;211:115-141.
5. Mendenhall MD, Hodge AE. Regulation of Cdc28 cyclin-dependent protein kinase
activity during the cell cycle of the yeast Saccharomyces cerevisiae. Microbiol. Mol.
Biol. Rev. 1998;62:1191-1243.