Romain Leroux 1, Jérémy Seurat 1, Antoine Croxo 1, Lucie Fayette 1, France Mentré 1
1 Iame / Inserm / Université Paris Cité (Paris, France)
Objectives
Several packages for design evaluation and optimisation in pharmacometrics are based on the Fisher Information Matrix (FIM). The were recently compared on several examples [1]. The R package PFIM has long implemented these methods and has been available on CRAN since 2022 [2]. PFIM 7.0, released in July 2025 (https://cran.r-project.org/web/packages/PFIM/index.html), introduced a complete rewrite of the package architecture using the R S7 object-oriented system [3], ensuring greater modularity, extensibility, and computational performance. However, two critical modelling features remained absent: discrete covariates (such as sex, treatment groups, or genotypes); and inter-occasion variability (IOV), which is essential for capturing within-subject variability across repeated occasions, particularly in crossover or multiple-dose designs.
PFIM 7.1 was developed to address these limitations with two primary objectives: (i) to implement FIM computation under four practical scenarios combining discrete covariates and IOV; and (ii) to introduce formal testing features for covariate effects, enabling researchers to compute expected power and the Number of Subjects Needed (NSN) to conclude on statistical significance (via comparison test), clinical relevance or non-relevance (known as equivalence for bioequivalence studies).
Methods
FIM evaluation is performed using a first-order linearisation of the NLMEM, resulting in a block-diagonal FIM. Individual parameters are modelled with between-subject random effects (IIV, variance-covariance matrix Ω), inter-occasion random effects (IOV, variance-covariance matrix Γ), and a vector β of discrete covariate effects. The core of PFIM 7.1 lies in the unified Covariate() class, which manages four distinct cases:
• Case 1 — standard population FIM with no covariates and no IOV;
• Case 2 — discrete covariates (binary or categorical, with user-defined frequencies and reference category) without IOV, where the population FIM is computed as a weighted sum of subgroup-level FIMs;
• Case 3 — discrete covariates with IOV for crossover trials, defined through treatment sequences and their proportions;
• Case 4 — mixed structures allowing for the simultaneous specification of covariates with and without IOV (e.g., sex and treatment crossover).
Regarding design optimisation, PFIM 7.1 preserves all legacy algorithms (Multiplicative, Fedorov-Wynn, Simplex, Particle Swarm Optimisation, Population Genetic-Based Optimisation). For each covariate effect β, the software now supports a statistical significance test (Wald test), a relevance test and a non-relevance test (TOST test) [4]. For the relevance and non-relevance, the expected power is by default computed at a significance level of α=0.05, with a default equivalence interval of [log(0.8), log(1.25)] aligned with international bioequivalence standards.
Results
The functionality of PFIM 7.1 was illustrated using a one-compartment oral PK model (ka, V, Cl) with IIV (30%), IOV (15%), and N=40 subjects [1]. Precision (SE, RSE) and the D-criterion were accurately computed for all parameters across the four covariate scenarios. Benchmarking against PopED [5] and mlxDesignEval [6] on these cases showed that RSE and population FIM values agreed to at least four decimal places (10−4 precision), confirming the numerical accuracy of the S7-based implementation [1].
For the two-period cross-over trial (Case 4), with sex effect on V and treatment effect on CL, the expected power to detect the significance of the sex effect on V was 41% with N=40, and the NSN to reach 90% power was 142. Given that the ratio for this effect was 1.2, thus close to the 1.25 boundary, the non-relevance power was only 10% with N=40. For the treatment effect on Cl (ratio 1.10), the expected power of non-relevance was 98% with N=40, requiring only 25 subjects to reach 90% power to conclude non-relevance (i.e. bioequivalence on AUC).
Conclusion
PFIM 7.1 extends the population design evaluation and optimisation framework in R by introducing discrete covariates, IOV, and their complex combinations within a modern S7 object system. The package allows researchers to assess the influence of covariate distributions and proportions on design performance, and the inclusion of tests and NSN calculations facilitates the design of trials aiming to conclude on statistical significance, clinical relevance, or clinical non-relevance. Current limitations include the lack of handling of continuous covariates and the absence of optimisation of the covariate distribution; these features are planned for future releases. Furthermore, the current implementation of IOV only handles repeated observation times between occasions, separated by a washout period. PFIM 7.1 is scheduled for publication on CRAN within the next few months.
References:
[1] Fayette L, Brendel K, Mentré F. Advances and further comparison of software tools for Fisher Information Matrix based design evaluation in pharmacometrics. Pharmaceutical Research, 2026
[2] Leroux R, Mentré F. PFIM: Population Fisher Information Matrix. R package version 7.0.2, https://github.com/packagePFIM/PFIM, http://www.pfim.biostat.fr/, 2025
[3] Vaughan D, Hester J, Kalinowski T, Landau W, Lawrence M, Maechler M, Tierney L, Wickham H. S7: An Object-Oriented System Meant to Become a Successor to S3 and S4. R package version 0.2.1.9000, 2025.
[4] Fayette L, Brendel K, Mentré F. 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.
[5] Nyberg J, Ueckert S, Stroemberg EA, Hennig S, Karlsson MO, Hooker AC. PopED: An extended, parallelized, nonlinear mixed effects models optimal design tool. Computer Methods and Programs in Biomedicine, 2012.
[6] Celliere G, Pierre M. mlxDesignEval: Design evaluation through calculation of relative standard errors (RSE) based on Monolix or Simulx projects. 2024; R package version 0.3.0, 2024.
Reference: PAGE 34 (2026) Abstr 12005 [www.page-meeting.org/?abstract=12005]
Poster: Methodology - Study Design