PK simulation and replicates

Dear NM users Happy new year! Is there a regulatory guideline or general rule/recommendation regarding the number of replicates to use in a popPK simulation and the impact on the distribution of the simulated variables? Thank you in advance for your help Kind regards Marie

Dear Marie Rajerson, The following is my opinion. 1) For simulations without inter-indivdual and intra-individual random variability: you would not need to worry about the number of replicates. 2) For simulations with inter-individual and intra-individual variability: 2-1) If you are doing a clinical trial simulation: The clinical trial design and group size you would like to evaluate decides the number of replicates. 2-2) If you are estimating the distribution of some variable (e.g., AUC), you can conduct power analysis for the number of replicates needed to achieve a certain level of precision for the central tendency and variability of that variable. However, a big enough number of replicates(e.g., 10000) suffices for precise estimation of the distribution, even without power analysis. Sincerely, Jun Seok Cha

Dear Marie, Someone else may comment on the regulatory guidelines, but before even getting into this here is something to think about. If you only intend to do simulations based on the point estimates of your population parameters (i.e. point estimates of THETA, OMEGA and SIGMA), then there is often no need to simulate a trial with replicates. For that case, just use a sufficient number of subjects for the statistic(s) you want to derive, or what you want to illustrate and simulate a single-replicate data set. One exception would be e.g. VPC, where you want to simulate replicate trials (all with the same set of population parameters) with the analysis data set, in order to establish CIs based on the data set at hand. This article may help you understand the number of replicates needed (Jonsson and Nyberg): https://pubmed.ncbi.nlm.nih.gov/35353958/ Similarly, if you want to also account for uncertainty in population parameters, the number of replicates needed depends on the confidence interval you want to report, and what precision you want to achieve. For example if you are interested in the group mean Ctrough,ss and also want to account for the uncertainty in this predicted value (by accounting for the uncertainty in population parameters): In general you would need more replicate trials in order to calculate the statistic with 99% CI as supposed to an 80% CI. And in all cases with replicate trials: you should calculate the statistic (e.g. a mean, or a percentile) separately for each replicate trial, not across all trials. I will stop there since I am not sure whether you are intend to simulate a single trial, or do the latter and account for the uncertainty as well. Best regards Jakob Jakob Ribbing, Ph.D. Principal Consultant & Client Operations Expert [email protected] +46(0)705-14 33 77 www.pharmetheus.com 

Dear Marie, There is a lot of confusion when it comes to replication in pop pk stochastic simulations. If you are referring to subjects then simulations of population predictions from your model will require a large sample size so that the sample statistic converges to the population parameter (e.g., sample mean from the simulations converges to the population mean as the sample size increases to infinity, this is known as the Law of Large Numbers in Statistics). If you want to quantify uncertainty in the population predictions via a confidence interval, then you will also want a sufficiently large number of replicate trials with the large sample size for each trial accounting for parameter uncertainty which can be thought of as trial-to-trial uncertainty. As Jacob indicates, the number of replicate trials needed increases with higher coverage probability for the CI. For a 90% CI, I typically go with 1000 replicated trials but if you wanted say a 99% CI you would want a considerably higher number of replicate trials. If the interest in the pop PK/PD stochastic simulations is to conduct clinical trial simulations for a proposed study design of a fixed (finite) sample size, then replicate trials accounting for the parameter uncertainty would be used to obtain prediction intervals (PI) for the sample statistic of the finite sample size of the proposed study design. The number of replicate trials for quantification of a PI like a CI will also depend on the selected coverage probability. Note for most study designs the fixed/finite sample size is usually too small for the sample statistic to converge to the population parameter and so prediction intervals essentially reflect sampling variation from one trial to the next as well as parameter uncertainty whereas confidence intervals do not because of the large (infinite) sample size the sample statistic will converge to the population parameter which is a fixed number, i.e., it has no sample-to-sample variation. One can think of a CI as a PI but with the sample size going to infinity. Thus, PIs are always wider than CIs because of the smaller sample size used in the simulations for each trial. In other words a PI will converge to a CI as the sample size gets larger. As Jacob indicates, the internal VPC is a special case where the prediction interval is not to make inference for a future trial but to assess the predictive performance of the current trial/data used to develop the model. In this setting the machinery for the stochastics simulations for VPCs is the same except we don't account for parameter uncertainty because we are not using these intervals to make inference for a future trial. The number of replicated trials of your observed data for the VPCs should be guided by the percentiles you want to summarize for these prediction (VPC) intervals. The more extreme percentiles you want to quantify the larger the number of replicate trials. Again, for VPC intervals constructed based on the 5th and 95th percentiles (i.e., inner 90% range) 1000 replicated trials would be reasonable. For more information about sample size and trial replication, and the distinction between CIs, PIs, and VPCs, see the following articles: Hu, C. "Variability and uncertainty: interpretation and usage of pharmacometrics simulations and intervals." JPP 2022;49:487-481. Kowalski, K.G. "Integration of Pharmacometric and Statistical Analyses Using Clinical Trial Simulations to Enhance Quantitative Decision Making in Clinical Drug Development." Stats in Biopharm Res 2019;11:85-103. Kind regards, Ken Kenneth G. Kowalski President Kowalski PMetrics Consulting, LLC Email: <mailto:[email protected]> [email protected] Cell: 248-207-5082