Well-tempered MCMC simulations for population pharmacokinetic models

Abstract

A full Bayesian statistical treatment of complex pharmacokinetic or pharmacodynamic models, in particular in a population context, gives access to powerful inference, including on model structure. Markov Chain Monte Carlo (MCMC) samplers are typically used to estimate the joint posterior parameter distribution of interest. Among MCMC samplers, the simulated tempering algorithm (TMCMC) has a number of advantages: it can sample from sharp multi-modal posteriors; it provides insight into identifiability issues useful for model simplification; it can be used to compute accurate Bayes factors for model choice; the simulated Markov chains mix quickly and have assured convergence in certain conditions. The main challenge when implementing this approach is to find an adequate scale of auxiliary inverse temperatures (perks) and associated scaling constants. We solved that problem by adaptive stochastic optimization and describe our implementation of TMCMC sampling in the GNU MCSim software. Once a grid of perks is obtained, it is easy to perform posterior-tempered MCMC sampling or likelihood-tempered MCMC (thermodynamic integration, which bridges the joint prior and the posterior parameter distributions, with assured convergence of a single sampling chain). We compare TMCMC to other samplers and demonstrate its efficient sampling of multi-modal posteriors and calculation of Bayes factors in two stylized case-studies and two realistic population pharmacokinetic inference problems, one of them involving a large PBPK model.

Keywords: Bayes factor; Bayesian inference; Thermodynamic integration; computational efficiency; physiologically-based pharmacokinetic model; population pharmacokinetics.

Figure 1.

Artificial data (black dots) and posterior fits (lines) of the linear regression model at the different perks (inverse temperatures). The solution is clearly bimodal.

Figure 2.

Two-compartment distribution model used for theophylline. Symbols are given in the text.

Figure 3.

Perks values sampled as a function of the TI MCMC sampler iteration in the case of the linear regression model.

Figure 4.

Three-dimensional plot of the positions of the 3997 parameter triplets (slope a, σ₁, σ₂) sampled from their joint target posterior distribution (perk = 1) in the parameter space of the linear regression model. They align along two thin needles and individual points are not discernable. Only a fraction (2.25%) of the prior parameter space is shown.

Figure 5.

Marginal target posterior distribution of the sampled values of slope a in the linear regression model. The two peaks are separated by a distance equal to about 600 times their width.

Figure 6.

Evolution of perks and associated pseudo-priors in the Gaussian inference problem at the various steps of the automatic perk scale adjustment. The scale was refined until perk zero was reached.

Figure 7.

Perks values sampled as a function of the TI MCMC sampler iteration in the Gaussian inference problem.

Figure 8.

Top: Histogram of the TI MCMC sampled values of mean μ, at perk 0, in the Gaussian inference problem, overlaid by the known prior density (red line). Bottom: log-density of the same sample (crosses) and of the prior (red line).

Figure 9.

Top: Histogram of the TI MCMC sampled values of mean μ, at perk 1, in the Gaussian inference problem, overlaid by the known posterior density (red line). Bottom: log-density of the same sample, normalized by the integrated likelihood (crosses) and of the exact posterior (red line).

Figure 10.

Perks values sampled as a function of the TI MCMC sampler iteration in the case of the theophylline two-compartment population PK model.

Figure 11.

Posterior fits of the two-compartment inter-individual variability population model to data on plasma theophylline concentration in six subjects (Trembath and Boobis 1980). Red line: maximum posterior predictions. Grey lines: 20 random posterior fits. Subjects (S1 to S6) are suffixed with the type of dosing: “i” means immediate release, “s” sustained release, “sr” sustained release repeated dosing, “srh” idem with a dose twice as high.

Figure 12.

Observed data values vs. maximum posterior inter-individual variability model predictions for the theophylline data set, in the case of either standard or TI MCMC simulations, with the two-compartment inter-individual variability model.

Figure 13.

Posterior fits of the two-compartment inter- and intra-individual variability population model to data on plasma theophylline concentration in six subjects (Trembath and Boobis 1980). Red line: maximum posterior predictions. Grey lines: 20 random posterior fits. Subjects (S1 to S6) are suffixed with the type of dosing: “i” means immediate release, “s” sustained release, “sr” sustained release repeated dosing, “srh” idem with a dose twice as high.

Figure 14.

Violin plots of theophylline posterior distributions of (A) population means μ, (B) inter-individual variances δ² and residual variance σ², for the two-compartment inter-individual variability model. For each parameter, the left sample (white) was obtained by TI MCMC simulations, and the right sample (gray) was obtained by standard MCMC simulations.

Figure 15.

Violin plots of theophylline individual parameters θ posterior distributions (subjects S1 to S6), for the two-compartment inter-individual variability model. For each subject, the parameters are respectively: k_r, k_a, V, k_e, k_cp, and k_pc. For each parameter, the left sample (white) was obtained by TI MCMC simulations, and the right sample (gray) was obtained by standard MCMC simulations.

Figure 16.

Standard MCMC vs. TI summaries (mean ± SD) of the posterior parameter distributions for the inter-individual variability model of theophylline pharmacokinetics. Colors code for population means (μ), population variances (Σ²), residual variance (σ²) and individual parameter values (θ_i). The non-matching parameter is Σ² for k_cp.

Figure 17.

Histograms of the posterior distribution of the inter-individual variance of k_cp, for the theophylline two-compartment inter-individual variability model. Panel A: parameter sample obtained by TI MCMC simulations; panel B: sample obtained by standard MCMC simulations.

Figure 18.

Violin plots of theophylline posterior distributions of the one-compartment model with inter- and intra-individual variability. Panel A: population means μ, inter-individual variances δ², intra-individual variances φ² and residual variance σ². Panel B: subjects’ means ξ. For each subject the parameters are respectively: k_r, k_a, V, and k_e. The parameter samples were obtained by TI MCMC simulations.

Figure 19.

Observed data values vs. maximum posterior model predictions for the acetaminophen data set, in the case of either standard or TI MCMC simulations.

Figure 20.

Violin plots of acetaminophen posterior parameter distributions. Panel A: population means μ. Panel B: inter-individual variances δ² and residual variances σ². For each parameter, the left sample (white) was obtained by TI MCMC simulations, and the right sample (gray) was obtained by standard MCMC simulations. Parameter indexing is specified in Supplementary Material Table S13.

Figure 21.

Violin plots of acetaminophen individual PBPK parameters θ posterior distributions (subjects S1 to S8). For each subject the parameters are in the same order as in Supplementary Material Table S13. For each parameter, the left sample (red) was obtained by TI MCMC simulations, and the right sample (gray) was obtained by standard MCMC simulations.