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| 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. | | The author is known as Araceli Gulledge. One of the things I love most is greeting card gathering but I don't have the time lately. For years she's been living in Kansas. Interviewing is what I do for a residing but I strategy on altering it.<br><br>Stop by my blog post :: car warranty ([http://Www.Tenx.com/UserProfile/tabid/62/userId/13166/Default.aspx similar site]) |
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| 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.
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| ==Statistical models==
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| For a mathematical definition of a dilution assay an observation space <math>U</math> is defined and a function <math>f:U\rightarrow\mathbb{R}</math> so that the responses <math>u\in U</math> are mapped to the set of real numbers. It is now assumed that a function <math>F</math> exists which relates the dose <math>z\in[0,\infty)</math> to the response | |
| :<math>f(u)=F(z)+e</math>
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| in which <math>e</math> is an error term with expectation 0. <math>F</math> is usually assumed to be [[continuous function|continuous]] and [[monotone function|monotone]]. In situations where a standard preparation is included it is furthermore assumed that the test preparation <math>T</math> behaves like a dilution (or concentration) of the standard <math>S</math>
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| :<math>F_{T}(z)=F_{S}(\rho z)</math>, for all <math>z</math>
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| where <math>\rho>0</math> is the relative potency of <math>T</math>. 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 <math>x=z^{\lambda}</math> with <math>\lambda>0</math> or a logarithmic transformation <math>x=\log (z)</math>. The latter can be shown to be a limit case of <math>\lambda\downarrow0</math> so if <math>\lambda=0</math> is written for the log transformation the above equation can be redefined as
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| :<math>F_{T}(x)=F_{S}(\rho^{\lambda}x)</math>, for all <math>x</math>.
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| Estimates <math>\hat F</math> of <math>F</math> 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|Generalized Linear Models]] with which a wide range of dilution assays can be modelled. Estimates of <math>F</math> may describe <math>F</math> satisfactorily over the range of doses tested, but they do not necessarily have to describe <math>F</math> beyond that range. However, this does not mean that dissimilar curves can be restricted to an interval where they happen to be similar.
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| In practice, <math>F</math> itself is rarely of interest. More of interest is an estimate of <math>\rho</math> 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.
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| Some special cases deserve particular mention because of their widespread use: If <math>F</math> is linear and <math>\lambda>0</math> this is known as a '''slope-ratio model'''. If <math>F</math> is linear and <math>\lambda=0</math> this is known as a '''parallel line model'''. Another commonly applied model is the [[probit model]] where <math>F</math> is the cumulative [[normal distribution]] function, <math>\lambda=0</math> and <math>e</math> follows a [[binomial distribution]].
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| ==Example: Microbiological assay of antibiotics==
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| [[image:DilutionAssay.png|Parallel line assay]]
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| 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 <math>u</math> is in this case the zone of inhibition and the diameter of this zone <math>f(u)</math> can be used as the measurable response. The doses <math>z</math> are transformed to logarithms <math>x=\log (z)</math> and the method of least squares is used to fit two parallel lines to the data. The horizontal distance <math>\log (\hat\rho)</math> between the two lines (shown in green) serves as an estimate of the potency <math>\rho</math> of the test preparation relative to the standard.
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| ==Software==
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| The major statistical software packages do not cover dilution assays although a statistician should not have difficulties to write suitable scripts or macros to that end. Several special purpose software packages for dilution assays exist.
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| ==References==
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| * Finney, D.J. (1971). Probit Analysis, 3rd Ed. Cambridge University Press, Cambridge. ISBN 0-521-08041-X
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| * Finney, D.J. (1978). Statistical Method in Biological Assay, 3rd Ed. Griffin, London. ISBN 0-02-844640-2
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| * Govindarajulu, Z. (2001). Statistical Techniques in Bioassay, 2nd revised and enlarged edition, Karger, New York. ISBN 3-8055-7119-4
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| ==External links==
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| '''Software for dilution assays:'''
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| *[http://www.bioassay.de PLA]
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| *[http://combistats.edqm.eu CombiStats]
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| *[http://www.unistat.com Unistat]
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| *[http://www.cambridgesoft.com/software/details/?ds=12&dsv=85 BioAssay]
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| [[Category:Pharmacology]]
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| [[Category:Biostatistics]]
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