The application and limitations of exposure multiplication factors in sublethal effect modelling

Thanks to growing interest and research in the field, toxicokinetic-toxicodynamic (TKTD) models are close to realising their potential in environmental risk assessment (ERA) of chemicals such as plant protection products. A fundamental application is to find a multiplicative scale factor which-when applied to an exposure profile-results in some specified effect relative to a control. The approach is similar to applying assessment factors to experimental results, common in regulatory frameworks. It also relies on the same core assumption: that increasing the scaling always produces more extreme effects. Unlike experimental approaches, TKTD models offer an opportunity to interrogate this assumption in a mathematically rigorous manner. For four well-known TKTD models we seek to prove that the approach guarantees a unique scale factor for any percentage effect. Somewhat surprisingly, certain model configurations may have multiple scale factors which result in the same percentage effect. These cases require a more cautious regulatory approach and generate open biological and mathematical questions. We provide examples of the violations and suggest how to deal with them. Mathematical proofs provide the strongest possible backing for TKTD modelling approaches in ERA, since the applicability of the models can be determined exactly.