Mathematical Description of the Experimental Results
In order to identify the type of dose-response relationship, experimental data for a particular outcome index should be approximated with an appropriate functional expression. However, the choice of approximating functions is not uniquely determined even where the dependence of the response on agent dose is monotonic. The problem becomes even more complex where the dependence is nonmonotonic.1,3
Monotonic dose-response relationships are often described with the Hill function (1), proportional to the cumulative function of the log-logistic distribution4-6
where b0,…,b3 are parameters determined with the least squares method by experimental data. Here, variable y represents the outcome and the variable x is the acting agent doses.
Another mathematical model for monotonic dependence presents a hyperbolic function (2) associated with the Michaelis-Menten equation, which is used, for example, to describe the rate of enzymatic reactions 7
Here, variable y also represents the outcome and the variable x is the acting agent doses.
However, often these functions are not enough to ensure good approximation of experimental data even in the case of monotonic dose-response dependence.
It is still more difficult to find a suitable mathematical model for a nonmonotonic dose-response relationship. Although there are lots of functional expressions for describing relationships of this type,1,3,8 none of them is universal, being confined to a certain limited area of research only.
At the same time, nonmonotonic relationship is often associated with manifestations of hormesis in cases where the effect is directed oppositely in 2 adjacent intervals of doses. This renders theoretical generalization of different variants of a nonmonotonic dose-response dependence a lot more complex problem since this type of dependence can feature more than 2 phases of this kind.1,2,9,10 The objective of approximating such multiphase relationships of a hormesis-associated type becomes a difficult challenge which requires using more complex power1,2 or special functions. 3
Like in the previous studies (for instance1,3,11 and many others), the type of combined cytotoxicity was estimated in a model based on the Response Surface Methodology (for example12,13). In this methodology, equation (3) describing the response surface
Y = Y (x1, x2) can be constructed by fitting its coefficients to experimental data
where Y is a quantitative effect (outcome) of a toxic exposure; x1 and x2 are the doses of the toxicants participating in the combination; f(x1,x2) is a regression equation with some numeric parameters. By virtually dissecting the response surface (3) at several model exposure levels, one obtains isoboles visually characterizing the type of two-factor combined action.
Today, the RSM is one of the most important general methods used in the analysis of combined effects produced by mixtures of bioactive substances, including toxic ones. This method enables the potentialities of effective experimental design to be used for approximating a response function. Constructing such approximation requires choosing an analytical model whose parameters would be determined by fitting to experimental data using the ordinary least squares method.
The quality of approximation by the proposed models was estimated by the usual and adjusted coefficient of determination. It turned out to be quite high for all models (both coefficients are at least .6). The evaluation of the statistical significance of the model parameters also shows their high significance (P < .001). In addition, it is obvious from the presented figures that the proposed models describe the observed experimental values well.
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