Population Pharmacodynamic Modeling Using the Sigmoid E-max Model: Influence of Inter-individual Variability on the Steepness of the Concentration-Effect Relationship. a Simulation Study

Johannes H. Proost*, Douglas J. Eleveld, Michel M. R. F. Struys

*Corresponding author for this work

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The relationship between the concentration of a drug and its pharmacological effect is often described by empirical mathematical models. We investigated the relationship between the steepness of the concentration-effect relationship and inter-individual variability (IIV) of the parameters of the sigmoid E-max model, using the similarity between the sigmoid E-max model and the cumulative log-normal distribution. In addition, it is investigated whether IIV in the model parameters can be estimated accurately by population modeling. Multiple data sets, consisting of 40 individuals with 4 binary observations in each individual, were simulated with varying values for the model parameters and their IIV. The data sets were analyzed using Excel Solver and NONMEM. An empirical equation (Eq. (11)) was derived describing the steepness of the population-predicted concentration-effect profile (gamma*) as a function of gamma and IIV in C50 and gamma, and was validated for both binary and continuous data. The tested study design is not suited to estimate the IIV in C50 and gamma with reasonable precision. Using a naive pooling procedure, the population estimates gamma* are significantly lower than the value of gamma used for simulation. The steepness of the population-predicted concentration-effect relationship (gamma*) is less than that of the individuals (gamma). Using gamma*, the population-predicted drug effect represents the drug effect, for binary data the probability of drug effect, at a given concentration for an arbitrary individual.

Original languageEnglish
Article number10
Number of pages10
JournalAaps journal
Issue number1
Publication statusPublished - 24-Jan-2021


  • pharmacokinetic-pharmacodynamic modeling
  • sigmoid Emax model
  • inter-individual variability
  • simulation

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