Structure-activity model, mathematical

At about the time of Free and Wilsons work, Kopecky and co-workers introduced a similar mathematical structure-activity model (34, 35). They tested four equations (Equations 8-11) for the expression of the quantitative difference between the log LD50 values of p- (34) and m-(35) disubstituted benzenes and benzene. 

Further, it is not at all strange that we can use number of carbon atoms to express the structure relation for many different properties. It is not being said that number of carbon atoms is in any way synonymous with solubility or boiling point or, in fact, that number of carbon atoms stands for solubility or boiling point. The structure-activity model essentially represents the relationship between structure and property in a quantitative mathematical form suitable for further use. 

Predictions based on structure-activity relationships (SARs), including qualitative and quantitative mathematical models, and the use of read-across data from related chemicals. 

In many cases of practical interest, no theoretically based mathematical equations exist for the relationships between x and y we sometimes know but often only assume that relationships exist. Examples are for instance modeling of the boiling point or the toxicity of chemical compounds by variables derived from the chemical structure (molecular descriptors). Investigation of quantitative structure-property or structure-activity relationships (QSPR/QSAR) by this approach requires multivariate calibration methods. For such purely empirical models—often with many variables—the... 

Keywords Skin permeability Percutaneous absorption Skin penetration Mathematical model Quantitative structure-activity relationships Permeability coefficient Human skin... 

There are several properties of a chemical that are related to exposure potential or overall reactivity for which structure-based predictive models are available. The relevant properties discussed here are bioaccumulation, oral, dermal, and inhalation bioavailability and reactivity. These prediction methods are based on a combination of in vitro assays and quantitative structure-activity relationships (QSARs) [3]. QSARs are simple, usually linear, mathematical models that use chemical structure descriptors to predict first-order physicochemical properties, such as water solubility. Other, similar models can then be constructed that use the first-order physicochemical properties to predict more complex properties, including those of interest here. Chemical descriptors are properties that can be calculated directly from a chemical structure graph and can include abstract quantities, such as connectivity indices, or more intuitive properties, such as dipole moment or total surface area. QSAR models are parameterized using training data from sets of chemicals for which both structure and chemical properties are known, and are validated against other (independent) sets of chemicals. 

Since one of the main aims of green chemistry is to reduce the use and/or production of toxic chemicals, it is important for practitioners to be able to make informed decisions about the inherent toxicity of a compound. Where sufficient ecotoxicological data have been generated and risk assessments performed, this can allow for the selection of less toxic options, such as in the case of some surfactants and solvents [94, 95]. When toxicological data are limited, for example, in the development of new pharmaceuticals (see Section 15.4.3) or other consumer products, there are several ways in which information available from other chemicals may be helpful to estimate effect measures for a compound where data are lacking. Of these, the most likely to be used are the structure-activity relationships (SARs, or QSARs when they are quantitative). These relationships are also used to predict chemical properties and behavior (see Chapter 16). There often are similarities in toxicity between chemicals that have related structures and/or functional subunits. Such relationships can be seen in the progressive change in toxicity and are described in QSARs. When several chemicals with similar structures have been tested, the measured effects can be mathematically related to chemical structure [96-98] and QSAR models used to predict the toxicity of substances with similar structure. Any new chemicals that have similar structures can then be assumed to elicit similar responses. 

In addition to using PMs, predictions of toxic hazard can also be made by using structure-activity relationships (SARs). A quantitative structure-activity relationship (QSAR) can be defined as any mathematical model for predicting biological activity from the structure or physicochemical properties of a chemical. In this chapter, the premodifer quantitative is used in accordance with the recommendation of Livingstone (1995) to indicate that a quantitative measure of chemical structure is used. In contrast, a SAR is simply a (qualitative) association between a specific molecular (sub)structure and biological activity. 

Structure-activity relationships (SARs) and quantitative structure-activity relationships (QSARs), referred to collectively as QSARs, can be used for the prediction of physicochemical properties, environmental fate parameters (e.g., accumulation and biodegradation), human health effects, and ecotoxicological effects. A SAR is a (qualitative) association between a chemical substructure and the potential of a chemical containing the substructure to exhibit a certain physical or biological effect. A QS AR is a mathematical model that relates a quantitative measure of chemical structure (e.g., a physicochemical property) to a physical property or to a biological effect (e.g., a toxicological endpoint). 

Quantitative Structure Activity Relationship (QSAR) is a method that makes predictions by the quantitative description of molecular properties with the use of descriptors of the chemical structure (Dearden 2003). This means QSAR models describe the quantitative or calculated relationship between a chemical structure and their biological activity (e.g. toxicity) with the help of chemical descriptors that are generated from the molecular structure (Durham and Pearl 2001). This relationship is described in from of a mathematical equation (e.g. log 1/C = a tt + b a +. .. + const). QSAR models generally show better predictivity if all compounds of a dataset involved in the prediction are derived from a congeneric series of compounds, that means they should all act by the same mechanism of action, since the physico-chemical and structural descriptors used in the QSAR reflect the same mechanism of action. Sometimes it is difficult to determine the mechanism of action, so series of compounds involved in a QSAR model are often restricted to a given chemical class in the hope that this will ensure a single mechanism of action (Dearden 2003). 

Hansch and Fujita used a Gaussian probability function to characterize the partitioning Step 1 and the Hammett function (log(k/k0) = pa) to describe the rate Step 2 in their model (10,12,13). By appropriate mathematical treatment, they arrived at the following general structure-activity relationship which has come to be termed the Hansch Equation. 

Benigni, R. and Giuliani, A. (1994). Quantitative Structure-Activity Relationship (QSAR) Studies in Genetic Toxicology. Mathematical Models and the Biological Activity Term of the Relationship. Mut.Res., 306,181-186. 

In studies of quantitative structure activity relationships (QSAR), the relative potencies of a series of drugs are subjected to analysis with the hope that biological potency will be described by a mathematical equation. QSAR is an actuarial or statistical method in which only objective data are used with no intrusion of models or mechanistic hypotheses. The equation that is obtained not only accounts for the relative potencies of the compounds, but from it are deduced predictions of the potencies of untested compounds if the equation is valid, the predictions are ineluctable. The method thus has the capacity of yielding new (structurally related) drugs with desired potency, perhaps drugs with enhanced selectivity or fewer side effects. 

A quantitative structure-activity relationship (QSAR) is a mathematical model used to establish an approximate relationship between a biological property of a compound and its structure-derived physicochemical and structural features [ ] The two main objectives of QSAR are to allow prediction of the biological properties of chemically characterized compounds that are not yet biologically tested and to obtain information on the molecular characteristics of a compound that are important for the biological properties. 

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