Contents lists available at ScienceDirect Toxicology Letters journal homepage: www.elsevier.com/locate/toxlet Dose–response analysis of toxicological and pharmacological mixtures with the model deviation ratio method: Problems and solutions Alina Macacu, Gilles Guillot* International Prevention Research Institute, Lyon, France ARTICLEINFO Keywords: Risk assessment Mixtures Dose response Concentration addition Loewe additivity Uncertainty Statistical test Statistical simulations R package Regulatory toxicology ABSTRACT Riskassessmentformixturesofchemicalsrequirestoinvestigatethemagnitudeoftheirpotentialadverseeffects onlivingorganisms.Thisisusuallydonebyassessinghowexperimentaltoxicologicalmixturedatadepartfrom the model of Loewe additivity. Several recent scientific studies propose to perform this task using an ad hoc method known as model deviation ratio (MDR) method. Moreover, the first official European regulatory document for the study of combined exposures explicitly recommends the use of the MDR method (EFSA Scientific Committee et al. Guidance on harmonised methodologies for human health, animal health and eco- logical risk assessment of combined exposure to multiple chemicals. EFSA Journal, 2019). WeshowherethattheMDRmethodisnotrootedinstatisticalprinciplesandcanleadtoerroneousclaims.We showhoweverthatthedistributionoftheMDRcanbeevaluatedbysimulationsandshowhowthisallowsusto deviseandcarryouta bonafide statisticaltest.Theproposedmethodaccountsforuncertaintyintheestimation ofED/EC50anddoesnotrequireaminimumsamplesize.Thecomputercodedeveloppedinthisstudyismade availableasanRpackagecalled MDR. 1. Background 1.1. Multiple exposures to chemicals and models for risk assessment Recent years have witnessed an increased awareness of multiple exposures to chemicals among international agencies (US EPA, 2007; Meek et al., 2009; World Health Organization, 2009; Environment, HealthandSafetyDivision–EnvironmentDirectorate–OECD,2018)as well as in the public opinion (ICF et al., 2019). This trend has been mirrored by a large research effort on toxicological mixtures (Bopp et al., 2018) and by the recent issue of a regulatory document by the European Food Safety Authority (EFSA) on the topic (EFSA Scientific Committee et al., 2019). An important aspect pointed out by recent research and regulatory documents is the need for models of the joint effectofchemicals.Thosemodelsallowscientiststoanalysedata,with the goals to identify mixtures of highest significance for public health but also to quantify the uncertainty attached to this assessment (Bopp etal.,2019). Towards this end, predominantly two models have been in use: Loewe additivity (Loewe, 1926) and Bliss independence (Bliss, 1939). The former assumes that compounds in a mixture act as if they were differentdilutionsofthesameactivesubstanceandisbestsuitedforthe modelling of compounds having the same mode of action. The latter assumes that compounds act statistically independently and is best suitedforthemodellingofcompoundswithdifferentmodesofactions, (see Cedergreen et al., 2008, for a recent discussion of those two con- cepts). Confronting experimental mixture data to those models allows scientiststoquantifythemagnitudeofdeparturebetweenobservedand predictedtoxicity.Inabsenceofsignificantdeparture,thesemodelswill be used for toxicity prediction. Assessing the significance of departure from those models is therefore of crucial importance for risk assess- ment.BothmodelsareusefulbutwenotethatLoeweadditivityisthe mostfrequentlyusedone.Forthesakeofconcisenessandwithoutloss of generality we focus here on Loewe additivity only. 1.2. The model deviation ratio method Confronting data to the Loewe additivity model involves assessing the significance of the departure between data and model prediction. Severalrecentscientificstudies(Beldenetal.,2007; CoorsandFrische, 2011; Coorsetal.,2012; Cedergreen,2014; Carnesecchietal.,2019)as well as the recent EFSA guidance (EFSA Scientific Committee et al., 2019) propose to perform this task using an ad hoc method known as modeldeviationratio(MDR)method.IntheMDRmethod,whichatthe best of our knowledge, was proposed first by Belden et al. (2007),the departurefromLoeweadditivityisdetectedbycomparinganestimate https://doi.org/10.1016/j.toxlet.2020.02.005 Received 19 November 2019; Received in revised form 20 January 2020; Accepted 11 February 2020 ⁎ Corresponding author. E-mail addresses: alina.macacu@i-pri.org (A. Macacu), gilles.b.guillot@i-pri.org (G. Guillot). Toxicology Letters 325 (2020) 62–66 Available online 25 February 2020 0378-4274/ © 2020 Elsevier B.V. All rights reserved. T