Limiting Dilution Assays for the Separation, Characterization, and Quantitation
of Biologically Active Particles and Their Clonal Progeny
Authored by Carl Taswell, published in Cell Separation: Methods and Selected Applications,
volume 4, chapter 6, pages 109-145, edited by Pretlow and Pretlow, 1987, Academic
Press.
Following quote excerpted from the introduction of the review chapter:
I. INTRODUCTION
Since the beginning of this century, limiting dilution assays (LDAs) have been used
to quantitate a wide variety of biologically active particles (BAPs) including bacteria
(Phelps, 1908), protozoa (Cunningham, 1915), viruses (Clark, 1927), tumor cells
(Hewitt, 1958), immunocompetent cells (Makinodan and Albright, 1962), and neurocompetent
cells (Barbarese et al., 1983). LDAs can also be used to quantify the effectiveness
of purification and depletion procedures (Taswell et al., 1979) and to separate
and characterize BAPs and their clonal progeny (Taswell et al., 1980). Recent articles
by Taswell (1981, 1984a,b) presented basic principles of LDAs, reviewed existing
methods, and introduced new methods for the problems of model discrimination, parameter
estimation, and design optimization. This article attempts to collect in one publication
all methods of statistical analysis relevant to LDAs and to present them in a unified
manner with a common terminology and notation.
Throughout most of their history, LDAs have been known as dilution assays, serial
dilution assays, dilution series, dilution tests, fermentation tube tests, coliform
density tests, etc., and limiting dilution analysis as the dilution method, dilution
series method, fermentation tube technique, multiple tube method, multitube fermentation
method, etc. It was only relatively recently that immunologists began using the
newer terms "limiting dilution assays" (Kennedy et al., 1966) and "limiting dilution
analysis" (Groves et al., 1970). Most bacteriologists, virologists, public health
officials, and sanitary engineers still use the older terms (Wilson, 1983; Greenberg
et al., 1985). Of all the different names for these bioassays, the term "limiting
dilution assay (LDA)" is the most descriptive, the most general, and therefore the
most appropriate for the collection of assays as a class. The words "limiting dilution"
emphasize two important and related aspects of this class: (1) the assays are based
on a process of dilution of the dose to extinction of the response, and (2) this
process requires that only the BAP to be quantitated is diluted to these limiting
doses while all other culture system constituents are provided at saturating (nonlimiting)
doses.
Assuming that certain fundamental hypotheses (Section I,B) are validated for each
case, the same methods of statistical analysis apply to all LDAs regardless of the
kind of BAP diluted in liquid suspension. Indeed, these methods also apply to procedures
used to quantitate BAPs that are not suspended in liquid. Botanists, ecologists,
and foresters quantitate plants on tracts of land; their observational studies analogous
to LDAs are called stocked-quadrat surveys (Blackman, 1935; Swindel, 1983). Agricultural
and veterinary scientists quantitate viruliferous insects in a vector population
capable of transmitting viral, bacterial, fungal, or parasitic diseases to plant
and animal hosts; their experimental procedures analogous to LDAs are apparently
not known by any particular name (Thompson, 1962; Kerr, 1971). All of these LDAs
and analogous procedures are dose–response assays that detect quantal responses
and require dilution of the dose to extinction of the response. They must be distinguished
from a related class of assays (such as plate, colony, plaque, and pock count assays)
that detect quantitative responses and do not require dilution to extinction. Different
methods of statistical analysis apply to this related but distinct class of assays
(Fisher et al., 1922; Roberts and Coote, 1965). These methods cannot be used for
LDAs.
A. Limiting Dilution Assays (LDAs)
LDAs detect binary (positive or negative) responses generated by BAPs in individual
in vivo or in vitro cultures within groups of replicate cultures that vary in the
dose of the test preparation from which the BAPs are sampled. LDAs can be used to
estimate the absolute number of BAPs [called the most probable number, MPN, or density
of coliform organisms by bacteriologists (Phelps, 1908; Wilson, 1983)], the 50%
endpoint on the dilution scale of the BAP test preparation [the dilution level at
which the group of replicates is 50% positive and 50% negative (Reed and Muench,
1938; Worcester, 1954)], and the relative frequency of BAPs [called the immunocompetent
cell frequency by immunologists (Taswell, 1981)]. These three parameter estimates
are obtained from two subclasses of LDAs: subclass I consists of all LDAs that can
be used to calculate absolute numbers and 50% endpoints but not relative frequencies,
and subclass II consists of all LDAs that can be used to calculate all three parameter
estimates. Absolute numbers are expressed as the number of BAPs per unit volume
of test preparation, 50% endpoints as units on the dilution scale of the BAP test
preparation, and relative frequencies as the proportion of BAPs within a mixture
of biologically active and inactive particles (BAPs and BIPs), Dilution levels for
50% endpoints may be necessary for drug or antisera titration studies but not appropriate
for any study where it is possible and meaningful to estimate the absolute number
of BAPs (because a dilution level is clearly less informative than an absolute number).
Therefore, they will not be considered further in this article.
The two subclasses can then be designated by their distinguishing parametric estimates
as absolute number LDAs (subclass I) and relative frequency LDAs (subclass II).
Both absolute number and relative frequency LDAs are biological assays for particles
of a specific type defined by their functional activity and called biologically
active or assayable particles (BAPs). Relative frequency LDAs, however, also incorporate
an accompanying physicochemical assay for particles of a general type defined by
their structural morphology or other physicochemical characteristics and called
physicochemically observable particles (POPs). As an example, LDAs are used to measure
the relative frequency of cytolytic T lymphocyte precursors (CTL-Ps) as the BAPs
within a mixture of leukocytes as the POPs (Taswell et al., 1979). In this example,
the functional activity of the BAPs is defined as cell differentiation and proliferation
producing a clone of cells that can kill target cells (assayed indirectly by 51Cr
release), while the structural morphology of the POPs is defined as standard leukocytic
morphology (observed directly by light microscopy). In absolute number LDAs, the
number of POPs (theoretically equal to the number of BAPs plus BIPs) is never known
because any physicochemical assay that could conceivably be used to observe them
is not performed due to impracticality or impossibility.
B. The Single-Hit Poisson Model (SHPM)
To analyze dose–response data from LDAs, it is necessary to validate a model incorporating
two fundamental hypotheses: one for the provision of the dose and the other for
the generation of the response. For the sampling of BAPs aliquoted to replicate
cultures, first McCrady (1915) assumed a binomial distribution hypothesis and then
Greenwood and Yule (1917) a Poisson distribution hypothesis. For the generation
of a ...
Following quote excerpted from the conclusion of the review chapter:
IX. CONCLUSION
LDAs were originally developed and have been most extensively used by public health
officials and sanitary engineers for the examination of water supplies, sewage and
waste water, and dairy products (Phelps, 1908; McCrady, 1915; Greenwood and Yule,
1917; Greenberg et al., 1985; Richardson, 1985). As discussed in Section I, LDAs
have also been used by investigators from many other biological and medical sciences.
It is the immunologists, however, who have been responsible for renewing interest
over the past decade in the continuing development of methods for the statistical
analysis of LDA data. This renewed interest derives from the increased size of assays
and complexity of applications in immunology. Sanitary engineers typically use 1,
5, or 10 replicates for each of from 1 to 3 dose groups to determine, for example,
whether the concentration of bacteria in drinking water does not exceed the maximum
safe level. Immunologists, however, typically use, say, 24, 60, 192, or more replicates
for each of from 3 to 6 or more dose groups to perform experimental comparisons,
clonal analyses, and partition analyses (Sections VI, VII, and VIII, respectively)
that are relatively much more complicated. Renewed interest in statistical research
for LDAs also derives from advances in computers and statistics. Efficient statistical
analysis of data from larger, more complicated assays would never have been practically
feasible without the assistance of the powerful yet economical personal computers
that have become available just within the past decade. Many new theories and methods
have been developed in statistics over the past several decades, some that have
been and some that have not yet been applied to LDAs, as discussed throughout this
article. Certainly, much work remains to be done.
This article attempts to provide an outline of all statistical methods relevant
to LDAs, reviewing past origins and recommending future directions. Apparently,
it is the first such attempt to collect statistical work on LDAs from many diverse
fields and to unify it with a common terminology and notation within a systematic
treatment of validity tests, parameter estimators, and assay design for both sample
and population LDAs. Hopefully, it will not be the last such attempt. The distinction
between sample and population LDAs (Section I,C) and the boundary between the concepts
of using sample LDAs to estimate sample parameters (Section III) and population
LDAs to estimate both sample and population parameters (Section IV,B) should be
explored further. These issues of parameter estimation should be investigated within
a conceptual framework that fully integrates model discrimination and selection
(Sections II and IV,A) and design optimization (Section V). The goal of this approach
should be to extract maximum information from past assays in a sequence in order
to obtain maximum information from future assays in the sequence. Furthermore, it
should also be to estimate the biological variance ... of the samples
... within the population in addition to the usual statistical variance ...
of the sample estimator ... given the sample ... A method for estimating ... [the
biological variance of the samples] ... is introduced
for the first time in this article (Section IV,B), and examples are provided with
estimates of the biological variance between samples within the same population
for several different populations (Section VI). Estimation of ... [the variance
between samples]... will enable investigators
to better characterize their study populations by quantitating the biological variation
between the BAP frequencies of test preparations from individual mice, patients,
or other sources. LDAs have been used for almost a century now. They have proved
to be valuable tools in the hands of biological and medical scientists for the separation,
characterization, and quantitation of BAPs and their clonal progeny. Continuing
development and proper use of new methods of statistical analysis for LDAs can only
serve to enhance the power of these tools.
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