P10: DesAna-MixToxExp

In project P10 (Design and analysis of mixture-toxicity experiments) we will develop methods for the design and analysis of experiments in which the effect of the combination of several toxic compounds is investigated. The goal of these experiments is to analyse if and by which degree several compounds act more or less than additive, see Bloch et al. (2023). In particular, it is of interest whether the combination of non-toxic concentrations remains non-toxic or whether it becomes toxic due to interaction effects.   

In the first part of the project, we will focus on the analysis and design of experiments, where two toxic compounds are combined. First, we will analyse the performance of established modelling approaches and extend them. In particular, we will fit data from mixture-toxicity experiments to parametric non-linear surface models, e.g. as described by Zhou et al. (2024). One pitfall of these models is that they usually assume monotonic increase, neglecting potential hormesis effects. We will thus develop non-linear surface models that can incorporate such non-monotonous effects. Moreover, we will develop new non-linear two-dimensional models by combining parametric one-dimensional concentration-response models using appropriate link functions.       

Another challenge of mixture-toxicity experiments with two compounds is their design, namely the determination of sets of concentrations considered within the experiment. Here, we will first determine the number of different combinations of concentrations required to obtain unique estimates of the parameters for the parametric two-dimensional models. In particular, we will analyse how many non-trivial combinations of the two compounds are needed for a unique fit, especially, when data on the individual concentration-response behaviour is already available. In a further step, we will improve the design by determining the optimal allocations of concentrations (and their combinations). On the one hand, we will calculate efficient designs based on the established D-optimality criterion, see Schorning et al. (2018), Papathanasiou et al. (2019) and Holland-Letz et al. (2020). On the other hand, we will develop new optimality criteria that are applicable when the goal of the experiment is to accurately estimate the threshold between toxic and non-toxic combinations. To achieve this, we will extend the concept of the minimum effective doses, which is established in the analysis of toxicity experiments with one compound. In particular, we will consider appropriate contour lines of the non-linear surfaces to define a minimum effective dose combination. We will then construct optimality criteria, which aim for the precise estimation of the minimum effective dose combination. Other optimality criteria will directly address the identification of potential synergistic effects, e.g., whether one of the compounds works as an enhancer of the other, see Bloch et al. (2023). Since the considered surfaces will be non-linear, any resulting optimality criteria will depend on the values of the parameters involved. Therefore, we will first investigate locally optimal designs, which are based on the assumption of explicit parameter values. Then, we will incorporate more robust versions of the optimality criteria, which are based on less informative knowledge about the parameters, see Schürmeyer et al. (2023). Finally, we will compare the performance of the different designs using both simulation studies and data from real mixture-toxicity experiments.     

In the second part of the project, we will consider the combination of more than two toxic compounds. First, we will investigate the scalability of the methods developed for the scenario of two compounds. This question is directly connected to the budget of the corresponding experiment, since the complexity of (non)-linear surface modelling approaches increases with the number of involved compounds. Consequently, it is only applicable if a sufficiently large number of data is available at different concentration combinations of all compounds. We will thus determine the minimal number of different concentration combinations required for the application of appropriate non-linear surface models. In the second step, we will consider the situation where the necessary number of different concentration combinations exceeds the budget for the mixture toxicity experiment. In this case, a unique fit to the non-linear surface model is no longer possible. Instead, we will develop optimality criteria and corresponding optimal designs that still allow the precise estimation of certain subsets of parameters of the model, e.g., the parameters that describe potential synergetic effects of the compounds under consideration.

• Bloch D, Diel P, Epe B, …, Hengstler JG (2023). Basic concepts of mixture toxicity and relevance for risk evaluation and regulation. Arch Toxicol, 97, 3005-3017. doi: 10.1007/s00204-023-03565-6
• Holland-Letz T, Leibner A, Kopp-Schneider A. (2020). Modeling dose-response functions for combination treatments with log-logistic or Weibull functions, Arch Toxicol., 94(1),197-204. doi: 10.1007/s00204-019-02631-2        
• Papathanasiou T, Strathe A, Overgaard RV, Lund TM, Hooker AC (2019). Optimizing Dose-Finding Studies for Drug Combinations Based on Exposure-Response Models, AAPS J, 21(5):95. doi: 10.1208/s12248-019-0365-3          
• Schorning K, Dette H, Kettelhake K, Möller T (2018). Optimal designs for non-competitive enzyme inhibition kinetic models, Statistics, 52:6, 1359-1378. doi: 10.1080/02331888.2018.1511716         
• Schürmeyer L, Schorning K, Rahnenführer J (2023). Designs for the simultaneous inference of concentration-response curves. BMC Bioinformatics, 24(1):393. doi: 10.1186/s12859-023-05526-3       
• Zhou Y, Sloan A, Menon S, Wang L (2024). Combination MCP-Mod for two-drug combination dose-ranging studies, Journal of Biopharmaceutical Statistics. doi: 10.1080/10543406.2024.23112544