Abstract
In this analysis, we first performed a critical review of one-compartment models used to describe metal toxicokinetics in invertebrates and found mathematical or conceptual errors in almost all published studies. In some publications, the models used do not represent the exact solution of the underlying one-compartment differential equations; others use unrealistic assumptions about constant background metal concentration and/or zero metal concentration in uncontaminated medium. Herein we present exact solutions of two differential-equation models, one describing simple two-stage toxicokinetics (metal toxicokinetic follows the experimental phases: the uptake phase and the decontamination phase) and another that can be applied for more complex three-stage patterns (toxicokinetic pattern does not follow two phases determined by an experimenter). Using two case studies for carabids exposed via food, based on previously published data, we discuss and compare our models to those originally used to analyze the data. Our conclusion is that when metal toxicokinetic follows a one-compartment model, the exact solution of a set of differential equations should be used. The proposed models allow assimilation and elimination rates to change between toxicokinetic stages, and the three-stage model is flexible enough to fit patterns that are more complex than the classic two-stage model can handle.
Highlights
One of the major challenges in assessing potential effects of toxicants to organisms, to efficiently counteract their negative impacts on the environment, is explicitly predicting the internal active concentration of toxic chemicals in the body and/or target organs
The one-compartment model with two stages Comparison of the original model by Jansen et al [4]; the one used by Kramarz [2]; and two versions of the exact solution of the differential equations 8 and 9: with (1) common kA and kE values for both phases of the experiment and (2) separate assimilation and elimination rates for each phase showed that the two models allowing the constants to differ between the phases gave a clearly better fit (Table 2, Fig. 1)
The exact solution with separate assimilation and elimination rates for the uptake and decontamination phases gave almost identical toxicokinetic parameters and R2adj as those obtained by Kramarz [2], except for kA2, which was not estimated by Kramarz (Table 2)
Summary
One of the major challenges in assessing potential effects of toxicants to organisms, to efficiently counteract their negative impacts on the environment, is explicitly predicting the internal active concentration of toxic chemicals in the body and/or target organs. Toxic effects estimated on the basis of internal body/tissue concentrations rather than on external exposure (e.g., concentration in food) are often far less variable among species, different chemicals with similar mode of action, and different environmental conditions [1]. In case of metals (which do not degrade like, e.g., pesticides) the simplistic toxicokinetic models are usually used for invertebrates. In such models, mathematical equations are fitted to experimental data on body burden as a function of time in an organism exposed to the contaminated medium. The models allow for estimating toxicant assimilation and elimination rates which can be used further for predictive simulations. Estimation of assimilation rate in the presence of simultaneous elimination is improved significantly if the uptake is followed by decontamination phase at which animals are offered uncontaminated food [5]
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