Dilution assay
Encyclopedia
The term dilution assay is generally used to designate a special type of bioassay
in which one or more preparations (e.g. a drug) are administered to experimental units at different dose levels inducing a measurable biological response. The dose levels are prepared by dilution in a diluent that is inert in respect of the response. The experimental units can for example be cell-cultures, tissues, organs or living animals. The biological response may be quantal (e.g. positive/negative) or quantitative (e.g. growth). The goal is to relate the response to the dose, usually by interpolation
techniques, and in many cases to express the potency/activity of the test preparation(s) relative to a standard of known potency/activity.
Dilution assays can be direct or indirect. In a direct dilution assay the amount of dose needed to produce a specific (fixed) response is measured, so that the dose is a stochastic variable defining the tolerance distribution. Conversely, in an indirect dilution assay the dose levels are administered at fixed dose levels, so that the response is a stochastic variable.
is defined and a function 
so that the responses 
are mapped to the set of real numbers. It is now assumed that a function 
exists which relates the dose 
to the response
in which
is an error term with expectation 0. 
is usually assumed to be continuous
and monotone. In situations where a standard preparation is included it is furthermore assumed that the test preparation
behaves like a dilution (or concentration) of the standard 
, for all 
where
is the relative potency of 
. This is the fundamental assumption of similarity of dose-response curves which is necessary for a meaningful and unambiguous definition of the relative potency. In many cases it is convenient to apply a power transformation 
with 
or a logarithmic transformation 
. The latter can be shown to be a limit case of 
so if 
is written for the log transformation the above equation can be redefined as
, for all 
.
Estimates
of 
are usually restricted to be member of a well-defined parametric family of functions, for example the family of linear function
s characterized by an intercept and a slope. Statistical techniques such as optimization by Maximum Likelihood
can be used to calculate estimates of the parameters. Of notable importance in this respect is the theory of Generalized Linear Models with which a wide range of dilution assays can be modelled. Estimates of
may describe 
satisfactorily over the range of doses tested, but they do not necessarily have to describe 
beyond that range. However, this does not mean that dissimilar curves can be restricted to an interval where they happen to be similar.
In practice,
itself is rarely of interest. More of interest is an estimate of 
or an estimate of the dose that induces a specific response. These estimates involve taking ratios of statistically dependent parameter estimates. Fieller's theorem
can be used to compute confidence intervals of these ratios.
Some special cases deserve particular mention because of their widespread use: If
is linear and 
this is known as a slope-ratio model. If 
is linear and 
this is known as a parallel line model. Another commonly applied model is the probit model
where
is the cumulative normal distribution function, 
and 
follows a binomial distribution.
An antibiotic
standard (shown in red) and test preparation (shown in blue) are applied at three dose levels to sensitive microorganism
s on a layer of agar
in petri dish
es. The stronger the dose the larger the zone of inhibition of growth of the microorganisms. The biological response
is in this case the zone of inhibition and the diameter of this zone 
can be used as the measurable response. The doses 
are transformed to logarithms 
and the method of least squares is used to fit two parallel lines to the data. The horizontal distance 
between the two lines (shown in green) serves as an estimate of the potency 
of the test preparation relative to the standard.
Bioassay
Bioassay , or biological standardization is a type of scientific experiment. Bioassays are typically conducted to measure the effects of a substance on a living organism and are essential in the development of new drugs and in monitoring environmental pollutants...
in which one or more preparations (e.g. a drug) are administered to experimental units at different dose levels inducing a measurable biological response. The dose levels are prepared by dilution in a diluent that is inert in respect of the response. The experimental units can for example be cell-cultures, tissues, organs or living animals. The biological response may be quantal (e.g. positive/negative) or quantitative (e.g. growth). The goal is to relate the response to the dose, usually by interpolation
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
techniques, and in many cases to express the potency/activity of the test preparation(s) relative to a standard of known potency/activity.
Dilution assays can be direct or indirect. In a direct dilution assay the amount of dose needed to produce a specific (fixed) response is measured, so that the dose is a stochastic variable defining the tolerance distribution. Conversely, in an indirect dilution assay the dose levels are administered at fixed dose levels, so that the response is a stochastic variable.
Statistical models
For a mathematical definition of a dilution assay an observation space is defined and a function so that the responses are mapped to the set of real numbers. It is now assumed that a function exists which relates the dose to the responsein which
is an error term with expectation 0. is usually assumed to be continuousContinuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
and monotone. In situations where a standard preparation is included it is furthermore assumed that the test preparation
behaves like a dilution (or concentration) of the standard , for all where
is the relative potency of . This is the fundamental assumption of similarity of dose-response curves which is necessary for a meaningful and unambiguous definition of the relative potency. In many cases it is convenient to apply a power transformation with or a logarithmic transformation . The latter can be shown to be a limit case of so if is written for the log transformation the above equation can be redefined as, for all .Estimates
of are usually restricted to be member of a well-defined parametric family of functions, for example the family of linear functionLinear function
In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
s characterized by an intercept and a slope. Statistical techniques such as optimization by Maximum Likelihood
Maximum likelihood
In statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....
can be used to calculate estimates of the parameters. Of notable importance in this respect is the theory of Generalized Linear Models with which a wide range of dilution assays can be modelled. Estimates of
may describe satisfactorily over the range of doses tested, but they do not necessarily have to describe beyond that range. However, this does not mean that dissimilar curves can be restricted to an interval where they happen to be similar.In practice,
itself is rarely of interest. More of interest is an estimate of or an estimate of the dose that induces a specific response. These estimates involve taking ratios of statistically dependent parameter estimates. Fieller's theoremFieller's theorem
In statistics, Fieller's theorem allows the calculation of a confidence interval for the ratio of two means.-Approximate confidence interval:...
can be used to compute confidence intervals of these ratios.
Some special cases deserve particular mention because of their widespread use: If
is linear and this is known as a slope-ratio model. If is linear and this is known as a parallel line model. Another commonly applied model is the probit modelProbit model
In statistics, a probit model is a type of regression where the dependent variable can only take two values, for example married or not married....
where
is the cumulative normal distribution function, and follows a binomial distribution.Example: Microbiological assay of antibiotics
An antibiotic
Antibiotic
An antibacterial is a compound or substance that kills or slows down the growth of bacteria.The term is often used synonymously with the term antibiotic; today, however, with increased knowledge of the causative agents of various infectious diseases, antibiotic has come to denote a broader range of...
standard (shown in red) and test preparation (shown in blue) are applied at three dose levels to sensitive microorganism
Microorganism
A microorganism or microbe is a microscopic organism that comprises either a single cell , cell clusters, or no cell at all...
s on a layer of agar
Agar
Agar or agar-agar is a gelatinous substance derived from a polysaccharide that accumulates in the cell walls of agarophyte red algae. Throughout history into modern times, agar has been chiefly used as an ingredient in desserts throughout Asia and also as a solid substrate to contain culture medium...
in petri dish
Petri dish
A Petri dish is a shallow glass or plastic cylindrical lidded dish that biologists use to culture cells or small moss plants. It was named after German bacteriologist Julius Richard Petri, who invented it when working as an assistant to Robert Koch...
es. The stronger the dose the larger the zone of inhibition of growth of the microorganisms. The biological response
is in this case the zone of inhibition and the diameter of this zone can be used as the measurable response. The doses are transformed to logarithms and the method of least squares is used to fit two parallel lines to the data. The horizontal distance between the two lines (shown in green) serves as an estimate of the potency of the test preparation relative to the standard.