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Design and analysis of dose-response experiments

Software available

In recent years, the DKFZ has put an increasing focus on the direct application of research results in concrete patient care. One of the topics is the development of individualized chemotherapeutic treatments based on tumor samples from different cancer types, for example in the context of the INFORM project. A major part of these projects is the cell culture based analysis of the susceptibility of the individual tumor samples to different available drugs. In this context, a large number of dose response experiments have to be performed, and one of the research interests of the biostatistics department is both the efficient planning and the proper analysis of these experiments.


  • Peterziel, H., Jamaladdin, N. et al.: Development of a functional patient-derived 3D multicellular platform for realtime personalized drug sensitivity profiling platform for real-time personalized drug sensitivity profiling of patient-derived 3D fresh tumor tissue cultures in the pediatric precision oncology program INFORM.To appear in: npj Precision Oncology, 2023
  • Weimer M, Jiang X, Ponta O, Stanzel S, Freyberger A., Kopp-Schneider A. (2012) The impact of data transformations on concentration-response modeling. Toxicology Letters 213, 292– 298.

Statistical optimal experimental design for dose response studies

In a dose response experiment, the substance at question usually has to be tested in increasing dose levels, with the goal of estimating a complete dose response function for any dose level. However, the ability to actually estimate the dose response relationship and the efficiency depends on the selection of which dose levels to test, and how to distribute observations among these dose levels. As this is a mathematical optimization problem, statistical optimal design theory can be used to find the best possible experimental designs. In the past, we derived these solutions for the concrete situations appearing in dose response trials, including trials with the aim of estimating the interaction between two substances given in conjunction. More recently, we focused on applying these solutions in practice. This includes both simplifying the designs to be more easily employed in practical situations, and by developing tools which allow non-mathematicians to investigate the efficiency of their experimental designs themselves. In this context, we developed a graphical representation illustrating how much information measurements under different dose levels contribute (see Fig.1). Furthermore, we created a web application using R shiny which allows end-users to create and evaluate experimental designs themselves.


Figure: A design heatmap. Both axes represent the potential log dose levels that could be used in an experiment. If a dose level on the x-axis crossed with the same dose level on the y-axis is colored red, this dose level will provide maximum information about a relevant subset of the dose response parameters and is thus a good candidate for the actual experiment. If a dose level on the x-axis crossed with a different dose level on the y-axis is colored red, both dose levels provide maximum information about the same subset of the parameters, and can thus be used interchangeably in the actual experiment. Design heatmaps are also featured in our WebApp.

  • Holland-Letz, T; Kopp-Schneider, A. Optimal experimental designs for dose-response studies with continuous endpoints. Arch. Tox. 89(11): 2059-2068
  • Holland-Letz, T; Kopp-Schneider, A. Optimal experimental designs for estimating the drug combination index in toxicology. Comp. Stat. Data Ana. 117: 182-193
  • Holland-Letz, T; Kopp-Schneider, A. The design heatmap: A simple visualization of D-optimality design problems. Biom. J. 62(8), 2013-2031
  • Holland-Letz, T; Kopp-Schneider, A. An R-shiny application to calculate optimal designs for single substance and interaction trials in dose response experiments. Tox. Lett. 337, 18-27
  • Holland-Letz, T. On the combination of c- and D-optimal designs: General approaches and applications in dose response studies. Biometrics 73(1): 206-213

Statistical analysis of dose response trials, especially for combination treatments

The statistical theory for the estimation of dose response curves is quite well developed, and usually proposes fitting log-logistic or Weibull type functions to experimental data. Practical challenges in this area however are how the curve fits can then be summarized into singular values (e.g. ED50/IC50, Drug Sensitivity Scores DSS) for easy comparisons of substances, especially taking into account that some of the substances might require different dose scales.
If the experiment however aims to investigate the combined effect of two different substances given in conjunction, the available theory is more limited. We extended the existing theory for estimating whether a substance combination has an effect which is larger or smaller than expected by a simple additive combination (so called synergistic or antagonistic interactions, "Loewe" additivity model). Towards this end, we developed mathematical formulas which predict the full dose response curve to be expected when a purely additive combination is assumed, thus allowing deviations to be more easily detected (Holland-Letz and Kopp-Schneider, 2020). Furthermore we again developed how the experimental design has to be chosen in order to estimate potential synergies at given combinations and dose levels with optimal precision.
Topics for future investigations will be how the interactions observed under specific combinations in an experiment can be extrapolated efficiently to dose combinations not yet used in an experiment, and how incorrect prior information about substances will affect the performance of experimental designs. .

  • Holland-Letz, T; Leibner, A; Kopp-Schneider, A: Modeling dose-response functions for combination treatments with log-logistic or Weibull functions. ARCHIVES OF TOXICOLOGY 94(1), 197-204, JAN 2020
  • Holland-Letz, T; Gunkel, N; Amtmann, E; Kopp-Schneider, A:Parametric modelling and optimal experimental designs for estimating isobolograms for drug interactions in toxicology. JOURNAL OF BIOPHARMACEUTICAL STATISTICS Volume: 28 Issue: 4 Pages: 763-777, NOV 2017

Statistical analysis for replicated experiments

Often, data from multiple experiments are available. In this context, it may be of interest to obtain a combined estimate for a parameter of interest, e.g. the EC50 or the Lowest Observed Effect Concentration (LOEC). LOEC is a parameter reflecting the responsiveness of an in vitro system to substances. It is estimated by reverse regression under, e.g., a log-logistic model. We have investigated the applicability of meta-analytic approaches for summarizing EC50 or LOEC estimates from repeated experiments, as well as of approaches for coping with potential non-normality and/or heteroscedasticity in the log-logistic model fit.

  • Jiang X, Kopp-Schneider A (2014) Summarizing EC50 estimates from multiple dose–response experiments: a comparison of a meta-analysis strategy to a mixed-effects model approach. Biom J 56(3):493–512
  • Jiang X, Kopp-Schneider A (2015) Statistical strategies for averaging EC50 from multiple dose–response experiments. Arch Toxicol 89(11):2119–2127
  • Calderazzo S. et al., Model-based estimation of lowest observed effect concentration from replicate experiments to identify potential biomarkers of in vitro neurotoxicity. ARCHIVES OF TOXICOLOGY 93(9): 2635-2644 (2019)

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