Using Delaunay triangulation and Voronoi tessellation to predict the toxicities of binary mixtures containing hormetic compound

Abstract

Concentration addition (CA) was proposed as a reasonable default approach for the ecological risk assessment of chemical mixtures. However, CA cannot predict the toxicity of mixture at some effect zones if not all components have definite effective concentrations at the given effect, such as some compounds induce hormesis. In this paper, we developed a new method for the toxicity prediction of various types of binary mixtures, an interpolation method based on the Delaunay triangulation (DT) and Voronoi tessellation (VT) as well as the training set of direct equipartition ray design (EquRay) mixtures, simply IDVequ. At first, the EquRay was employed to design the basic concentration compositions of five binary mixture rays. The toxic effects of single components and mixture rays at different times and various concentrations were determined by the time-dependent microplate toxicity analysis. Secondly, the concentration-toxicity data of the pure components and various mixture rays were acted as a training set. The DT triangles and VT polygons were constructed by various vertices of concentrations in the training set. The toxicities of unknown mixtures were predicted by the linear interpolation and natural neighbor interpolation of vertices. The IDVequ successfully predicted the toxicities of various types of binary mixtures.

Novel Segmented Concentration Addition Method to Predict Mixture Hormesis of Chlortetracycline Hydrochloride and Oxytetracycline Hydrochloride to Aliivibrio fischeri

“…This method achieved good prediction for mixture U-CRC with the same effect sign (positive or negative), but the prediction for mixture J-CRC with both positive and negative effect cannot be completely satisfied. Qu et al [35] developed an interpolation method based on the Delaunay triangulation and Voronoi tessellation to predict mixture hormesis.…”

JSFit: a method for the fitting and prediction of J- and S-shaped concentration–response curves

“…For example, for the model coefficient min, the description of the minimum inhibitory ratio of hormetic CRCs is more than 0 sometimes. 22 Although, the M5 model has such shortcomings, the fitting quality of M5 to the hormetic concentration–effect data is acceptable, which is the reason why it is so widely used.…”

“…The concentration–response/effect data set ( C – E ) used in this paper comes from the literature. 22 The data set consists of toxic effects and luminescence inhibition ratios (each three replications) of 1-ethylpyridinium chloride ([epy]Cl) at 12 concentration levels in seven exposure times (0.25, 2, 4, 6, 8, 10 and 12 h) on Vibrio Qinghaiensis sp.-Q67 (Q67) (Table S1 in the ESI † ). In the set, the CRCs of [epy]Cl in the first 0.25 h and 2 h are S-shaped, while those in the latter five times (4, 6, 8, 10 and 12 h) are J-shaped (Fig.…”

US EPA: Is there room to open a new window for evaluating potential sub-threshold effects and ecological risks?

“…Hormetic responses have also been induced by various mixtures, creating the need to predict sub-NOAEL/hormetic effects of mixtures (Belz and Piepho, 2017;Ge et al, 2020;Liess et al, 2019;Ma et al, 2020;Mo et al, 2020;Qu et al, 2019a,b;Tang et al, 2019;Wang et al, 2020;Zhu et al, 2019). Given potential limitations of CA and IA models to predict mixtures hormesis (Qu et al, 2019b), contemporary mathematical approaches considering hormesis/sub-NOAEL effects of binary mixtures have also been developed, including interpolation methods based on (i) the Delaunay triangulation and Voronoi tessellation and the training set of direct equipartition ray design (EquRay) mixtures (Qu et al, 2017) and (ii) Similarity and Triangulation (SimTri) (Qu et al, 2019c). More recently, an inverse distance weighted (IDW) interpolation method has been proposed for accurate prediction of multi-component mixtures, considering also hormetic components (Qu et al, 2019b).…”