2023 - A Coruña - Spain

PAGE 2023: Methodology - New Modelling Approaches
Robin Chaux

Meta-analysis of parameter estimates of non-linear mixed effect models using Approximate Bayesian Computation: application to population pharmacokinetics

Robin Chaux (1,2), Paul Jacques Zufferey (1,2), Edouard Ollier (1,2)

(1) Inserm U1059 Dysfonction Vasculaire et Hémostase (2) Unité de Recherche Clinique, CHU Saint-Etienne, Hôpital Nord, Saint-Etienne, France


Non-Linear Mixed Effect Models (NLMEM) are a gold standard tool to analyze pharmacokinetic (PK) data. Generally, numerous population PK models are published for a specific drug (1). These models often differ in terms of structural model, which can be problematic when it comes to aggregating multiple models into a single meta-model.

An ideal approach would be to perform a meta-analysis using individual patient’s data (IPD)(2). But the limited availability of IPD makes the realization of IPD meta-analysis challenging.

A common solution is to conduct a meta-analysis based on aggregated data, for example by pooling mean estimates across studies for each parameter. But characteristics of published models are heterogeneous, and can bias results.

The development of a statistical procedure that combines parameter estimates from NLMEM analysis by considering the respective structure of each model is required. But performing such estimation using likelihood-based methods is challenging. One solution could be to use a likelihood-free approach such as Approximate Bayesian Computation (ABC)(3). This method allows inference of parameters from complex statistical models where the true likelihood function is unknown.

This work proposes to apply the ABC framework to perform a meta-analysis of parameters estimates of NLMEM from multiple studies.

Objectives :

  • Develop an ABC procedure for the meta-analysis of parameter estimates from multiple NLMEM without access to IPD
  • Estimate the performances of this ABC procedure combined with a post-processing method


First, the structure of a reference PK model, inferred from the literature, must be chosen by the user. A distribution of prior for each parameter for this reference model has to be defined.

The ABC procedure then consists of the constitution of a data base of virtual study results reproducing the data found in the literature. The simulation process is composed of 4 successive steps: 

1) Sample a set of parameters for the reference model from the prior distributions

2) Simulate virtual individual PK data for each study based on their own design, using the reference model and the parameters sampled in step 1

3) Estimate the parameters for each of the PK models selected in the literature, using the simulated individual PK data in step 2

4) Store the parameter estimates obtained in step 3, and the parameter values for the reference model sampled in step 1 

For each simulated result, a distance from the results published for each model is calculated. Then, the posterior distribution of the reference model parameters can be approximated by taking only the reference model parameters associated with a distance lower than a certain threshold. 

An additional post-processing step based on the regression method (4) was also applied to improve estimation accuracy. 

The proposed ABC method was evaluated using dummy data. ABC parameters estimations were compared to i) their “true” values (used to generate observed dummy study results), ii) estimations obtained from an IPD meta-analysis and iii) univariate aggregated data meta-analysis. 


11 dummy population PK analysis results were simulated with various designs and model structures. The reference PK model was 2 compartmental, with random effects on all parameters. 

We illustrated the results with estimations obtained for the central volume of distribution (Vc) and its between-subject standard deviation (sdVc). The “true” Vc value was set to 6.4 L. With the IPD analysis, Vc estimation was 6.44 L, 95% CI (Confidence Interval) = [6.11 : 6.77]. With aggregated data meta-analysis, the estimation was 7.52 L, 95% CI = [6.18 : 8.86]. Using ABC with post-processing, the estimation was 5.77 L, 95% CI = [4.60 : 6.87].

For sdVc, the “true” value was set to 0.45. With IPD analysis, the estimation was 0.44, 95% CI = [0.41 : 0.48]. With aggregated data meta-analysis, the estimation was 0.12, 95% CI = [0.02 : 0.21]. Using the ABC with post-processing, the estimation was 0.41, 95% CI = [0.19 : 0.60].

The proposed method will also be applied using real data extracted from the population PK studies of tranexamic acid.


The proposed ABC method allows to meta-analyze parameter estimates of multiple NLMEM from the literature by taking into account the specific structures of each models. Estimations were more precise and less biased than with simple aggregation methods where the model structure was not considered.

[1] Aljutayli A, Marsot A, Nekka F. An Update on Population Pharmacokinetic Analyses of Vancomycin, Part I: In Adults. Clin Pharmacokinet. juin 2020;59(6):671-98.
[2] Riley RD, Lambert PC, Abo-Zaid G. Meta-analysis of individual participant data: rationale, conduct, and reporting. BMJ. 31 août 2010;340(feb05 1):c221-c221.
[3]  Sisson SA, Fan Y, Beaumont MA, éditeurs. Handbook of Approximate Bayesian Computation / edited by Scott A. Sisson, Yanan Fan, Mark A. Beaumont. Boca Raton: CRC Press; 2020.
[4] Raynal L, Marin JM, Pudlo P, Ribatet M, Robert CP, Estoup A. ABC random forests for Bayesian parameter inference. Bioinforma Oxf Engl. 15 mai 2019;35(10):1720-8.

Reference: PAGE 31 (2023) Abstr 10720 [www.page-meeting.org/?abstract=10720]
Oral: Methodology - New Modelling Approaches