Lucie Fayette 1,2, Karl Brendel 2, France Mentré 1
1 Université Paris Cité and Université Sorbonne Paris Nord, Inserm, IAME,F-75018 Paris, France (, ), 2 Pharmacometrics, Ipsen Innovation, Paris, France (, )
Objectives
One aim of pharmacometrics is to explain inter-individual variability (IIV) through covariate identification [1]. Covariate effects are assessed through parameter ratios and confidence intervals [2], with clinical relevance concluded when the 90% CI lies outside a predefined equivalence interval. Designing studies with sufficient precision is therefore essential and optimal design strategies are recommended [3]. They rely on Fisher Information Matrix (FIM), which has no analytical solution for non-linear mixed effects models (NLMEM), and maximization of criteria such as D-criterion.
For continuous response, first-order linearization of the NLMEM yields a Gaussian approximation with an analytical FIM [4], suitable for design evaluation and optimisation [5], and implemented in several software tools [6]. Nonetheless, existing implementations neither handle continuous covariates nor compute power for relevance/non-relevance tests. The impact of design and covariate distribution on these tests and on the number of subjects needed (NSN) to reach desired power remains unexplored. Furthermore, despite recommendations to cover the full covariate range [3, 7], no quantitative framework exists to optimise covariate allocation in future trials, yet evidence of relevance/non-relevance in specific sub-populations is required for dose adjustment and labelling.
Beyond continuous-response, FIM computation based on Monte Carlo integration over observations combined with Markov Chains Monte-Carlo [8] (MCMC) or Adaptive Gaussian Quadrature [9] (AGQ) to integrate over random effects has been proposed for non continuous outcomes but never extended to joint models linking continuous and time-to-event responses.
We therefore pursued two objectives:
1.Extend linearised FIM computation for continuous response models to continuous covariates, derive relevance/non-relevance power and optimise allocation of covariates among subjects to be included in future trials
2.Develop optimal design strategies for joint models linking continuous and time-to-event responses
Methods
1.Continuous responses
FIM computation was extended to models with continuous covariates by calculating its expectation over the joint distribution of the covariates. This integral was approximated either with Monte-Carlo sampling [10] or with a faster and deterministic Gauss-Legendre quadrature (GLQ) combined with copula modelling [11]. Assuming known model, a given design and covariate effects, CI on ratios and power of clinical relevance/non-relevance tests were derived from FIM [10]. Predictions were compared to clinical trial simulation (CTS).
For design optimisation, the domain of continuous covariates was segmented into clinically meaningful intervals and their proportions optimised, along with the proportion of each category for discrete covariates. The resulting convex optimisation problem with linear constraints was solved using Projected Gradient Descent (PGD) [11].
This approach was applied to a one-compartment IV bolus PK model with linear elimination including Body mass index (BMI) and Sex effects on volume and creatinine clearance (CLCR) on clearance to reflect renal function (RF), with a single-dose design and samples at 1, 4 and 12 h. Initial covariate distribution was imported from NHANES [12]. BMI and CLCR were segmented into standard clinical intervals. Sensitivity analysis assessed varying CLCR effect, IIV magnitude and various constraint settings to better reflect real-life conditions.
2.Joint models
FIM computation using MC integration combined with MCMC or AGQ was extended to joint models linking continuous and time-to-event responses. Accuracy was evaluated against CTS on a tumor growth–survival model [13]. Sum of Lesion Diameter (SLD) dynamics followed a Stein-Fojo model with lactate dehydrogenase (LDH) status (binary) affecting baseline SLD, and Alkaline Phosphatase (ALP) influencing baseline and tumour growth rate. Survival was modelled by a Weibull hazard with tumour growth rate as link and two binary covariates: Sex and LDH. The design included 250 subjects with rich SLD sampling (every 8 weeks until week 24, then every 12 weeks until death) and 3-year follow-up. Hyper-parameters to compute FIM were tuned to ensure convergence and D-criterion precision below 5%. Design influence was studied by varying follow-up duration (1/2/3 years), sampling (rich vs sparse, keeping one observation out of two) and sample size. Finally, influence of covariate distribution was investigated by optimising the proportion of the four Sex × LDH combinations using PGD.
Results
1.Continuous responses
FIM computation with continuous covariates was validated against CTS [10] and enabled quantifying how the power of relevance/non-relevance tests decreases with reduced sample size, fewer observations, or higher IIV.
Optimal distribution often placed zero weight on intermediate intervals, favouring extreme covariate intervals. Sensitivity analysis showed an overall stable shape for optimal distributions across varying IIV and covariate effect values. With constraints ensuring at least 5% representation per continuous interval and a 10% cap on Severe RF interval, the NSN to achieve 80% power on Sex and CLCR relevance and BMI non-relevance decreased from 602 to 230 (>60% reduction).
2.Joint models
Both MC combined with MCMC and AGQ were shown to accurately predict uncertainty for joint model linking continuous and time-to-event responses. AGQ was fastest, reducing computation time by 100 compared with CTS (200 datasets), enabling exhaustive design comparisons and covariate distribution optimisation.
In our example, longitudinal‑parameter uncertainty was barely affected by follow-up duration or sampling richness, whereas survival‑parameter uncertainty decreased substantially from 1-year to 2-year follow-up. Interestingly, although NSN to achieve below 15% uncertainty in the link parameter decreased with longer follow-up and richer sampling, NSN was very similar between a 2-year follow-up with rich sampling and a 3-year follow-up with sparse sampling.
Optimal covariate distributions were highly stable across designs and consistently required at least 40% of subjects with high LDH. Optimal distribution always increased the power to detect LDH relevance on both baseline SLD and survival. Moreover, implementing the optimal distribution in a sparse 2-year follow-up design yielded higher powers than a rich 3-year follow-up design with the initial distribution.
Conclusion
We introduced MC and quadrature methods to compute FIM for continuous, discrete, and joint models, including continuous and discrete covariates. These methods accurately predict uncertainty on covariate effects and power of clinical relevance or non-relevance tests, enabling assessment of designs before clinical trials and computation of NSN to reach target power. We further proposed an efficient framework to optimise covariate allocation across sub populations, substantially reducing NSN and providing group specific NSN targets relevant for dose adjustment and labelling. The methods are implemented in open-source R packages, with planned extensions to broaden accessibility.
References:
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10. Fayette, Brendel & Mentré. Using Fisher Information Matrix to predict uncertainty in covariate effects and power to detect their relevance in Non-Linear Mixed Effect Models in pharmacometrics. Journal of Pharmacokinetics and Pharmacodynamics (2025).
11. Fayette, Brendel & Mentré. Optimising covariate allocation at design stage using Fisher Information Matrix for Non-Linear Mixed Effects Models in pharmacometrics. medRxiv (2025).
12. National Health and Nutrition Examination Survey Data. Hyattsville, MD: U.S. Department of Health and Human Services, Centers for Disease Control and Prevention 2009-2020.
13. Alvares & Mercier. Bridging the gap between two-stage and joint models: The case of tumor growth inhibition and overall survival models. Statistics in Medicine (2024)
Reference: PAGE 34 (2026) Abstr 12115 [www.page-meeting.org/?abstract=12115]
Poster: Oral: Lewis Sheiner