What are conditions that binomial distribution approaches to Poisson distribution?

Counting Cells Scientifically: The Poisson Distribution

Before 1900, statistical methods were not rigorously applied in either clinical medicine or experimental science. The journalBiometrika began publication in 1901. In 1907, an article by “Student” demonstrated that the theoretical minimum error of cell counts with a hemocytometer varies with the square root of the number of cells actually counted,8 fitting the Poisson distribution described 70 years earlier. Poisson statistics apply to cells counted by any method and to other objects encountered [and counted] in cytometry, notably the photoelectrons generated by scattered or emitted light from cells interacting with cytometers' detectors. William S. Gossett had published under the pseudonym “Student” because his employers at the Guinness brewery were concerned that their competitors might benefit as they had from statistics [and cytometry]; he had counted yeasts rather than blood cells in the hemocytometer.

In 1910, Ronald Ross, who had won the 1902 Nobel Prize in Medicine and would soon be knighted for his discovery that mosquitoes transmitted malaria, applied Gossett's findings to calculate how much blood he and his fellow malariologists needed to analyze to detect small numbers of parasites with reasonable precision. The required amount, several microliters, spread thickly on a glass slide, would take an observer over an hour to examine thoroughly using a high-power oil immersion lens. This might be acceptable for research but would be difficult to implement on a regular basis for clinical use; there was, however, no technology even imaginable as a replacement for a human observer at that time.

Counting cells using a hemocytometer and a microscope requires only that the observer be able to distinguish the cells of interest from everything else in the sample. Even that level of discrimination may not always be necessary. Consider the cellular ecology of human blood, a common sample for cytometry.

Red blood cells are the most abundant [~5,000,000/µL whole blood]; their very numbers require a sample to be diluted several hundredfold to keep cells separated enough to be counted. The RBC concentration in whole blood is calculated from the number counted and the known dilution factor. Normal RBC volume is approximately 90 fL.

The typical WBC concentration in normal blood is 5000 to 10,000/µL, meaning that only one or two WBCs accompany each 1000 RBCs. WBCs vary in size from approximately 200 fL [lymphocytes] to more than 500 fL [monocytes], but there are larger lymphocytes and smaller monocytes. Although their hemoglobin content, lack of a nucleus, and smaller size make RBCs simple to discriminate from WBCs by microscopy or cytometry, most modern automated cell counters, which simply measure approximate cell size, do not make the distinction and instead include WBCs in RBC counts, with negligible effects on accuracy.

It had been known since the early days of hemocytometry that RBCs could be lysed and WBCs preserved for counting by diluting a blood sample with a hypotonic medium or with chemicals such as acids or detergents, and the same dilution procedure was later adapted to flow cytometric counters, which typically count WBCs in blood diluted approximately 1:10.

Poisson Events Described

The Poisson distribution arises from situations in which there is a large number of opportunities for the event under scrutiny to occur but a small chance that it will occur on any one trial. The number of cases of bubonic plague would follow Poisson: a large number of patients can be found with chills, fever, tender enlarged lymph nodes, and restless confusion, but the chance of the syndrome being plague is extremely small for any randomly chosen patient. This distribution is named for Siméon Denis Poisson, who published the theory in 1837. The classic use of Poisson was in predicting the number of deaths of Prussian army officers from horse kicks from 1875 to 1894; there was a large number of kicks, but the chance of death from a randomly chosen kick was small.

Conventional Radiation

In order to achieve durable tumor control, all or at least a significant proportion of clonogenic cells must be eliminated so that they can no longer maintain the tumor. The probability of tumor control is derived from the Poisson distribution using the equationP =e−n, whereP is the probability of tumor control andn is the average number of survival clonogens after radiation. For example, reduction of a clonogenic population consisting of initially 109 cells by at least 9 logarithms would give a probability of 37% tumor control, and reduction by 10 logarithms would give about a 90% probability of tumor control.

Relationship to the Binomial Distribution

The Poisson distribution approximates the binomial distribution closely when n is very large and p is very small. It is the limiting form of the binomial distribution when n→∞, p→0, and np = μ are constant and