ReviewA mathematical approach to study combined effects of toxicants in vitro: Evaluation of the Bliss independence criterion and the Loewe additivity model
Introduction
Organisms can be exposed to a mixture of different toxicants in the environment and although the toxicity of the single compounds might be well known, their simultaneous presence might induce non-overlapping toxic effects. Therefore, studies of interactions among toxicants are of fundamental interest and practical importance in toxicological sciences. The term “additivity” is used when several (two or more) compounds act without any interaction among them and the total effect does not differ from what can be expected from the dose-effect relations of the individual agents. The terms “antagonism” and “synergism” are used when there is an interaction among toxic compounds, i.e. when the total effect is lower (antagonism) or higher (synergism) than expected (Groten et al., 2001). In molecular terms, the toxicity of a compound depends among other things on its affinity to target sites at cellular level and it might be decreased or increased by the presence of other toxic substances that biologically modify cellular conformation and expression, sometimes affecting the cellular defense system and detoxification capability.
Only in a few cases is the exact toxicological mechanisms of a compound perfectly known and represented by a definite binding site. In fact, for most toxicants there are numerous potential target sites and even less is known about possible interactions. One way forward in elucidating the effects of combined exposure is by mathematical models. Different mathematical approximations have been introduced to define a “non-interaction” surface in an n-dimension space and different basic assumptions have been done to justify the use of models that take into account the simultaneous presence of several toxic compounds inside a cell, and more in general, inside an organism. There is no final agreement on the definition of agent interaction. Some reviews have been written about the subject and some aspects of the problem are still debated, with a particular regard to the biological plausibility of the different mathematical approaches (Greco et al., 1995, Suhnel, 1998). A recent review by Chou discusses possible different interaction models with several examples in vivo and in vitro (Chou, 2006).
The aim of this paper is to discuss the proposed models in the literature and to show some possible mathematical approaches for in vitro co-exposure studies that might be particularly useful when the toxicological pathway(s) of the toxic substances are partially or not known.
Section snippets
Bliss independence criterion and Loewe additivity model
The most commonly used models to study combined effects of substances in vivo and in vitro are the Bliss independence criterion and the Loewe additivity model (Greco et al., 1992).
(1) The main assumption of the Bliss independence criterion is that two or more toxic agents act independently from one another (Greco et al., 1995, Bliss, 1939, Berenbaum, 1981). In other words, if fulfilling the criterion, the mode, and possibly also the site of action of the compounds in the mixture, always differ.
In vitro models and the pure dose–response curves: the Hill function
The Hill function was introduced by Hill, 1910 to explain the binding between hemoglobin and oxygen. This function has been extensively used in toxicology to extrapolate the dose–response curves of single toxicants (reviewed in Greco et al., 1995, DeLean et al., 1978, Poch and Pancheva, 1995). This function has the advantage that, contrarily to other used functions (sigmoidal functions, polynomial, etc.), it can represent a biochemical simplification of interactions between toxic compounds and
The Hill function and combination experiments
The Hill function has been used as a pure curve model in several in vitro and in vivo combination experiments, analyzing two or more toxic substances (reviewed in Greco et al., 1995, Dressler et al., 1999, White et al., 2003, Jonker et al., 2005). In particular, authors have tested two curves with the same n and different (or equal) EC50s, but also models with a variable n. Since n is a parameter bound to the slope of the curve, it is sometimes called “the Hill slope”, also indicated as h. Some
The Hill function and combination experiments: the surface of non-interaction
If in an in vitro experiment an Effect (E), or preservation of a state, for example viability or survival (V = 1 − E), is due to simultaneous presence of some toxicants and is a function of toxicant concentration, it is possible to write functions that describe the non-interaction curve for two or more combined toxic substances (E = f(x1, … ,xn)), both for the Bliss independence criterion and the Loewe additivity model.
Here, we show functions with two toxicants (x, y) using the Hill function to describe
Statistical analysis of the surface of non-interaction
We assume a case where the experimental value of a parameter at fixed concentrations of x and y in combination (±SD) is known. How can we compare this experimental value to the theoretical dose–response curve and say that the effect is significantly synergistic/antagonistic or only additive?
One possibility is to use the model of Greco et al., 1995 and specific software that calculate the significance of the α value introduced by him with modifications to Eq. (11) (Greco et al., 1995, Greco et
The limitations of isobolographic methods: EC10 and EC50 in a simulation
In Goldoni et al., 2003, the limitations of the exclusive use of EC50 to test the toxicity of a compound in vitro have been shown. In brief, EC50 says nothing about the early toxicity of a compound and it is very difficult to compare EC50 in vitro with some parameters of early toxicity calculated in vivo (NOAEL – no observed adverse effect level, LOAEL – lowest observed adverse effect level, BMD – benchmark dose). So, we proposed the use of a Benchmark dose (coincident in our case with EC10,
Deviations between non-interaction surfaces calculated with the Bliss independence criterion and the Loewe additivity model
To better comprehend the deviations that can be found with the use of the two models, we simulated an experiment with two toxic substances, whose dose–response pure curves are represented by the equations:where in all combination experiments, for simplicity, n was maintained constant for both substances. With n values of 0.2, 1, 2, 5, we used the Eqs. (9), (10) to calculate the non-interaction surfaces with the Bliss and Loewe methods, combining seven
Conclusions
At this point, it is still unclear what model is optimal to study combined effects of toxicants in vitro, in particular if the pure dose–response curves are steep. As observed in the last paragraph, the choice of a model is a major concern, because at some co-exposure concentrations the differences in outcome are dramatic. In Table 1, the main pros and cons of each method are presented, based on the discussion in Section 2 and the mathematical simulations performed in the Sections 4 The Hill
Conflict of interest statement
All authors declare that they have no competing interests.
Acknowledgement
Supported by EC (Contract number: FOOD-CT-2003-506143).
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