Modeling dose–response functions for combination treatments with log-logistic or Weibull functions

  • Tim Holland-LetzEmail author
  • Alexander Leibner
  • Annette Kopp-Schneider
In silico


Dose–response curves of new substances in toxicology and related areas are commonly fitted using log-logistic functions. In more advanced studies, an additional interest is often how these substances will behave when applied in combination with a second substance. Here, an essential question for both design and analysis of these combination experiments is whether the resulting dose–response function will still be a member of the class of log-logistic functions, and, if so, what function parameters will result for the combined substances. Different scenarios might be considered in regard to whether a true interaction between the substances is expected, or whether the combination will simply be additive. In this paper, it is shown that the resulting function will in general not be a log-logistic function, but can be approximated very closely with one. Parameters for this approximation can be predicted from the parameters of both ingredients. Furthermore, some simple interaction structures can still be represented with a single log-logistic function. The approach can also be applied to Weibull-type dose–response functions, and similar results are obtained. Finally, the results were applied to a real data set obtained from cell culture experiments involving two cancer treatments, and the dose–response curve of a combination treatment was predicted from the properties of the singular substances.


Combination index Dose–response studies Loewe additivity Weibull function Log-logistic function 



The authors would like to thank Darell Doty Bigner for the friendly gift of the D425 Med cell line.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.German Cancer Research CenterHeidelbergGermany
  2. 2.Hopp Children’s Cancer Center Heidelberg (KiTZ)HeidelbergGermany
  3. 3.Center for Child and Adolescent MedicineHeidelberg University HospitalHeidelbergGermany

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