Implementing a Bayesian approach using Stan with Torsten: Population pharmacokinetics analysis of somatrogon

Abstract

Fully Bayesian approaches are not commonly implemented for population pharmacokinetic (PK) modeling. In this paper, we evaluate the use of Stan with R and Torsten for population PK modeling of somatrogon, a recombinant long‐acting growth hormone approved for the treatment of growth hormone deficiency. As a software for Bayesian inference, Stan provides an easy way to conduct MCMC sampling for a wide range of models with efficient sampling algorithms, and there are several diagnostic tools to evaluate the MCMC convergence and other potential issues. Three different sets of priors were evaluated for estimation and prediction: a weakly informative uniform set, a moderately informative set, and a very informative set of priors. All three prior sets showed good performance and all chains mixed well. There were some minor differences in the final parameter posterior distributions while implementing different prior sets, but the posterior predictions covered the observations nicely, not only for the individuals included in posterior sampling but also for new individuals. The impact of a centered versus non‐centered parameterization were evaluated, with the non‐centered approach improving the estimation time, but it was still computationally intensive. Computational resources had the biggest impact on sampling time. Stan took approximately 2.5 h total for the MCMC sampling on a high‐performance computing platform (6 cores) and may be reduced further with additional computational resources. The model and comparisons presented show that with adequate computational resources, the Bayesian approaches using Stan and Torsten are useful for population PK analysis, especially for the analysis of special populations, small sample datasets, and when complex model structures are needed.

Study Highlights.

• WHAT IS THE CURRENT KNOWLEDGE ON THE TOPIC?

While Bayesian analyses can be advantageous in dealing with small data set or sparse collected samples, the use of R, Stan, and Torsten for Bayesian population PK (PopPK) modeling has not been widely utilized.

• WHAT QUESTION DID THIS STUDY ADDRESS?

This study addresses the benefits and limitations of using Stan for PopPK modeling and compares the approach to a standard frequentist approach with a rare disease example where limited data is available.

• WHAT DOES THIS STUDY ADD TO OUR KNOWLEDGE?

This study provides insights into the usage of R, Stan, and Torsten for PopPK modeling. Stan is advantageous for its efficient MCMC sampling algorithm, and it can add values to frequentist approaches by implementing prior information to help the parameter posterior sampling. With adequate computational resources, MCMC sampling could be completed within an acceptable time even when the PK samples were collected over a long time period.

• HOW MIGHT THIS CHANGE DRUG DISCOVERY, DEVELOPMENT, AND/OR THERAPEUTICS?

Bayesian approaches for PopPK modeling using R, Stan and Torsten can be valuable when limited data is available and flexibility in the model specification is required.

INTRODUCTION

Growth hormone deficiency is a rare disease caused by the insufficient secretion of growth hormone by the pituitary gland. It can be congenital or acquired. ¹ Somatrogon [NGENLA™] is a recombinant growth hormone, approved for the treatment of growth hormone deficiency (GHD) in children and adolescents from 3 years of age with a once‐weekly subcutaneous injection. The pharmacokinetics (PK) of somatrogon has been assessed in children and adults with GHD. ² In children, after doses of 0.25–0.66 mg/kg, peak concentrations were observed 6–12 h post dose. The peak concentration after the 0.66 mg/kg dose was 1151 ng/mL and the AUC_inf was 28,085 ng·h/mL. The estimated half‐life was 22.4 h. ³

A 2‐compartment population PK (PopPK) model with a delayed first‐order absorption, combined with the covariate effects of body weight with allometric scaling, was previously developed using the first‐order conditional estimation with interaction (FOCEI) method in NONMEM with data from two Phase II (Study CP‐4‐003, N = 54 adults; Study CP‐4‐004, N = 42 pediatrics) studies (“Phase II PopPK model”). The PK parameters and the covariate effects were estimated with separate residual errors for the time period prior to 6 months and the time period after 6 months. To assess the effect of anti‐drug antibody on clearance, the 2‐compartment PopPK model was updated with the effect of time varying anti‐drug antibody status at each observation (ADAT) on apparent clearance in a subsequent analysis using the Phase III (Study CP‐4‐006, N = 109 pediatrics) pediatric participants (“Phase III PopPK model”), with the values of most other PK parameters being fixed to the final estimates from the Phase II PopPK model in the update. ⁴ While fixing parameters to evaluate one specific covariate effect is not uncommon, a major limitation of this approach is that uncertainty in the parameter estimates is ignored, which will impact the uncertainty in the predictions made using the model.

Bayesian estimation for PopPK modeling is much less common than standard frequentist approaches ⁵ for several reasons. Standard software such as NONMEM and Monolix provide straightforward estimation algorithms (e.g., FOCEI, SAEM in NONMEM or SAEM in Monolix), which have generally shorter run times than fully Bayesian approaches that require a computationally intensive iterative process (Markov chain Monte Carlo [MCMC]). The $PRIOR subroutine in NONMEM with method FOCEI could be used for more hybrid approaches where a frequentist approach was used to initially, or previously, to develop a model followed by the $PRIOR subroutine to facilitate Bayesian inferences. ⁶ , ⁷ One major advantage of implementing a fully Bayesian analysis is the posterior sampling via MCMC as it allows for model flexibilities and provides a natural and intuitive way to quantify uncertainties in the parameter estimates and predictions. ⁸ A common application of Bayesian PopPK modeling is to extrapolate to new indications or patient populations. In the aforementioned scenarios, the final estimates from a previous model are commonly used when specifying the priors for the new or updated model. ⁶ , ⁹

Stan is a probabilistic programming language, which provides Bayesian statistical inference with Markov chain Monte Carlo (MCMC) sampling. ¹⁰ Stan uses advanced MCMC algorithms for efficiently exploring complex posterior distributions and supports parallelization, which can speed up the computation of posterior samples. Stan also provides a flexible and expressive modeling, and it provides several diagnostic tools for assessing the convergence of the MCMC chains and diagnosing potential problems. Torsten is a package for Stan that provides efficient functions to solve pharmacokinetic/pharmacodynamic (PK/PD) models. ¹¹ , ¹² , ¹³ The PopPK analyses presented here were conducted using Stan through R with the Torsten package.

As predictions were previously made from the earlier developed PopPK model, the purpose of the analysis presented here was to evaluate a Bayesian approach to PopPK modeling using Stan and R with the Torsten package. Of particular interest was the impact of the selection of the prior distributions for each of the parameters and the effect on the precision of the parameter estimates and the subsequent predictions as well as the estimation time. The objectives of this work were as follows: (1) to implement a Bayesian approach for PopPK modeling of somatrogon using pediatric participants from a Phase II and a Phase III study considering several different approaches for prior distributions (non‐informative vs. informative); (2) to use the results from the model to make posterior predictions for pediatric participants in a Phase III study; (3) to compare the sampled posterior distributions via MCMC with the previous estimates from FOCEI method in NONMEM; and (4) to assess the computation requirements for using the Torsten's analytical solution. This includes evaluating the impact of centered versus non‐centered parameterizations on estimate time. All together, the results from this work will provide some additional insight into the benefits and challenges of Bayesian PopPK modeling in Stan as well as a potential use cases where the Bayesian approach should be considered or even preferred to a frequentist approach.

METHODS

Ethical conduct of studies

The final protocols and informed consent documentation for the studies included in this analysis were reviewed and approved by an institutional review board. The parents or guardians of all participants provided signed and dated informed consent. All studies were compliant with the ethical principles of the Declaration of Helsinki and all International Conference on Harmonisation Good Clinical Practice guidelines.

Clinical trial data

There were three clinical trials used for these analyses. The first was a Phase II study CP‐4‐003 in adult patients. ¹⁴ The second was a Phase II study CP‐4‐004 in pediatric patients. ³ The third was a Phase III study CP‐4‐006, which was also in pediatric patients. ¹⁵ They are herein referred to as Studies 003, 004, and 006, respectively. The nominal timing of the PK samples available from the pediatric patients in the clinical studies is shown in Table 1. These three studies were used in two previous analyses (Table 2). The demographic summary information of the participants related to this Bayesian analysis is shown in Table 3.

TABLE 1.

PK sample summary.

Study (Protocol)	Nominal time after dosing (h)
	0	6	12	24	48	72	96	120	168	Other
Phase II CP‐4‐004
All observations a	119	16	12	29	34	271	406	19	13	7
Observations up to 1 year after first dose b	56	16	12	26	33	148	239	18	11	1
Phase III CP‐4‐006
All Observations a	12	0	3	7	41	319	183	29	218	14
Observations up to 1 year after first dose b	12	0	3	7	39	298	173	29	206	8

^a All observations were used in previous PopPK analysis and used in the NONMEM $Prior Subroutine model in this analysis.

^b Observations up to 1 year after first dose were used in Stan models in this analysis. Observations in CP‐4‐004 was used to fit the Bayesian PopPK model, and CP‐4‐006 was used to show the model's ability in predicting new data.

TABLE 2.

Analyses and studies.

Analysis	CP‐4‐003 (Phase II)	CP‐4‐004 (Phase II)	CP‐4‐006 (Phase II)	Comments
	N = 54, adult	N = 42, pediatric	N = 109, pediatric
Phase II PopPK model	√	√
Phase III PopPK model			√	Fix most of the parameters to the estimates from Phase II PopPK model
Stan models		√	Used as an external data set to test the model prediction ability	Use estimates from Phase II and Phase III PopPK models as priors
NONMEM with $PRIOR		√		Use estimates from Phase II and Phase III PopPK models as priors

TABLE 3.

Demographic and dose summary.

Covariates and dose summary		Study (Protocol)
		Phase II (004)	Phase III (006)
	Total N	42	109
Body weight (kg)	Median (min–max)	14.8 (10–26.3)	19.3 (8–46.1)
	Mean ± SD	16.2 ± 4.5	19.7 ± 7.1
Age (year)	Median (min–max)	5.5 (3–11)	7.9 (3–12)
	Mean ± SD	6 ± 2.2	7.8 ± 2.7
Creatinine CL (mL/min)	Median (min–max)	42.7 (27.6–66.8)	48.9 (22–93.3)
	Mean ± SD	44 ± 8.9	49.7 ± 15.3
Sex; N (%)	Male	28 (66.7)	82 (75.2)
	Female	14 (33.3)	27 (24.8)
Race; N (%)	White	40 (95.2)	81 (74.3)
	Asian	0 (0)	24 (22)
	African American	1 (2.4)	–
	American Indian or Alaska Native	–	1 (0.9)
	Other or Missing	1 (2.4)	3 (2.8)
ADA Status a ; N (%)	Negative	24 (57.1)	24 (22.0)
	Positive	18 (42.9)	85 (78.0)
ADA Status a (within 1st year); N (%)	Negative	32 (76.2)	24 (22.0)
	Positive	10 (23.8)	85 (78.0)
Dose (mg/week)	Median (min–max)	6.74 (3.05–17.5)	13.5 (0.6–32)
	Mean ± SD	7.65 ± 3.71	14.2 ± 4.9
Dose (mg/kg/week)	Median (min–max)	0.482 (0.228–0.711)	0.657 (0.065–0.762)
	Mean ± SD	0.47 ± 0.165	0.649 ± 0.036

^a Participants were considered as “ADA positive” if they had at least one ADA‐positive test result during the study period or the 1st year after the first treatment.

Data inclusion criteria

All participants in study 004 who took at least one dose and had at least one PK sample while on treatment were included in the final posterior distribution sampling for parameters. All participants in studies 004 and 006 (new data) who took at least one dose and had at least one PK sample while on treatment were included in the posterior prediction. To speed up the MCMC estimation, only PK samples collected within 1 year of the study start were included in the analysis.

Modeling approach

In this Bayesian PopPK model, the plan was to estimate all PK parameters and covariate effects (including the effect of time varying ADA status on apparent clearance) using the pediatric participants from study 004. Posterior prediction sampling was then conducted for the pediatric participants in the studies 004 and 006 to evaluate the predictive performance of the model. ³ The parameterization of the 2‐compartment structural model as well as parameterizations for all fixed and random effect parameters followed the same approach used in the models developed using NONMEM with method of first‐order conditional estimation with interaction (FOCEI). Allometric scaling, adjusted for body weight, was used for apparent clearance (CL/F), apparent intercompartmental clearance (Q/F), apparent volume of distribution in the central compartment (Vc/F), and apparent peripheral volume of distribution (Vp/F).

The effect of ADAT (dichotomized as positive or negative and time‐varying) on the apparent clearance of the drug was estimated using a proportional parameterization as shown below, where ADAT may take value of 1 (positive) or 0 (negative).

$$P_i=P_{\mathrm{pop}}·\left(1+{\theta }_{\text{ADAT}}·\text{ADAT}\right)$$

The interindividual variability (IIV) in four of the PK parameters: apparent clearance CL/F, apparent central volume of distribution V_c/F, first‐order absorption rate constant Ka, and an additional between‐patient variance on the apparent clearance CL/F was considered if the observation was with positive anti‐drug antibody status (ADAT = 1). These were all modeled using a multiplicative exponential random effect. The random effects, ${\eta }_{\mathit{ik}}$, were assumed to be multivariate normally distributed with mean 0 and covariance matrix $\mathrm{\Omega }$. Although the random effects were assumed to be independent in the previous PopPK modeling, a full covariance matrix was estimated here allowing for correlation between the random effects. An ${\eta }_{\mathit{ik}}$ represents the interindividual random effect that accounted for the ith individual's deviation from the population value of each of the k = 4 parameters listed above.

Residual variability was estimated using an additive residual error structure with log‐transformed concentrations as the dependent variable,

$$\log \left(Y_{i,j}\right)=\log \left(F_{i,j}\right)+{ϵ}_{i,j}$$

where $Y_{i,j}$ was the observed concentration value for the ith individual at time $j$, $F_{i,j}$ was the model predicted individual concentration in participant $i$ at time $j$, ${ϵ}_{i,j}$ was the normally distributed intraindividual (residual) random effect with a mean of zero and variance ${\sigma }^2$.

Three different sets of priors were evaluated to understand the effect of priors on the MCMC‐sampled values, convergence, and posterior distribution. Bounded uniform distributions were applied to each parameter as the weakly informative prior set; normal distributions that were centered at the parameter estimates from the Phase II and Phase III PopPK models were applied as the moderate‐informative prior set and the very informative prior set. The only difference between the moderate‐informative prior and the very informative prior was that the variability was larger in the moderate‐informative prior set than the very informative prior. The parameter ranges for all the priors were restricted to plausible values (e.g., CL/F cannot take negative values) with upper or/and lower bounds declared in the Stan model file. To prevent the MCMC sampling from converging to a solution with flip flop kinetics, the absorption rate constant (ka) was defined with a lower bound of λ₁ (the overall rate constant associated with the distribution phase of a two‐compartment model). A Wishart prior distribution was used for all the covariance parameters. A scaled LKJ prior could be the option if a more informative prior for covariance parameters is desired since it permits flexible specification for each IIV parameter. Table 4 shows the details of the priors for each parameter.

TABLE 4.

Priors set up.

	Previous NONMEM model estimates	Uniform prior set	Moderate informative prior set c	Very informative prior set c
CL/F (L/h)	0.472 a	U (0, 5)	N (0.472, 2)	N (0.472, 0.5)
Q/F (L/h)	0.014 a	U (0, 5)	N (0.0135, 0.1)	N (0.0135, 0.05)
Vc/F (L)	10.9 a	U (0, 20)	N (10.9, 3)	N (10.9, 1)
Vp/R (L)	2.37 a	U (0, 20)	N (2.37, 3)	N (2.37, 1)
Ka (1/h)	0.313 a	U (0, 5)	N (0.313, 1)	N (0.313, 0.5)
$\sigma$	0.55 a	U (0, 5)	N (0.55, 1)	N (0.55, 0.5)
Lag time (h)	0.353 a	U (0, 5)	N (0.353, 1)	N (0.353, 0.5)
Weight effect on CL/F and Q/F	1.26 a	U (0, 5)	N (1.26, 3)	N (1.26, 1)
Weight effect on Vc/F and Vp/F	1.74 a	U (0, 5)	N (1.74, 3)	N (1.74, 1)
Proportional Change in Apparent Clearance for Positive ADA Status (ADAT effect on CL/F)	−0.258 b	U (−1, 5)	N (−0.258, 3)	N (−0.258, 1)
Variances on CL/F, Vc/F, Ka, and variance of ADAT effect on CL/F	–	Wishart (5, diag (0.1, 0.1, 0.1, 0.1))	Wishart (5, diag (0.1, 0.1, 0.1, 0.1))	Wishart (5, diag (0.1, 0.1, 0.1, 0.1))

Abbreviations: ADAT, Time‐Varying Anti‐drug Antibody Status at each observation; CL/F, apparent clearance from central compartment; Ka, absorption rate constant; Q/F, apparent clearance between peripheral and central compartment; Vc/F, apparent central compartment volume; Vp/F, apparent peripheral compartment volume.

^a Estimates using NONMEM 2‐cpt model with study CP‐4‐003 (adult) and study CP‐4‐004 (pediatric) (Phase II PopPK model).

^b Estimates using NONMEM 2‐cpt model with study CP‐4‐006 (pediatric) (Phase III PopPK model).

^c All PopPK parameters were declared with proper lower bounds (−1 for ADAT effect on CL/F and 0 for the rest), resulting in truncated normal priors.

Six MCMC chains with 500 warm‐up iterations and 500 sampling iterations were used to estimate the posterior distributions with each prior set. PK concentrations were predicted for pediatric patients from both the Phase II (Study 004) and the Phase III (Study 006, not used in model estimation) studies from the posterior distributions of each of the parameter. The 90% posterior prediction intervals and the posterior median of the PK concentrations were calculated for each pediatric participant and compared with the observed PK samples for that participant.

One of the widely used metrics to evaluate convergence of the MCMC chains is $\widehat{R}$, “Rhat,” which compares the between‐ and within‐chain estimates for each of the model parameters. Higher $\widehat{R}$ values indicate poorer mixing of the chains. Gelman and Rubin recommended that the samples are used only if all $\widehat{R}$ values are less than 1.05, ¹⁶ while Vehtari et al. recommended 1.01 as a more stringent threshold. ¹⁷ Additionally, the bulk and tail effective sample sizes (ESS) are measures of the sampling efficiency for the location and the tails of the posterior distributions, respectively. They are presented with the $\widehat{R}$ values and the caterpillar mixing plots.

Evaluation of parameterization on sampling time

The MCMC sampling time and sampling efficiency can vary depending on whether a centered or non‐centered parameterization is utilized. ¹⁸ We implemented a semi‐centered parameterization for individual PK parameters in this study (see script in Data [Link], [Link], [Link], [Link]) where the IIV effects were sampled from the multivariate‐normal (0, OMEGA) distribution and the individual post hoc PK parameters were assembled using the sampled parameter centers (THETAs) and the sampled IIV effects in “transformed parameters” block.

Additionally, we tested each of the model/prior sets with a centered parameterization where the individual PKs were sampled from the multi‐normal (sampled parameter center, sampled OMEGA) distribution, as well as a non‐centered parameterization where the IIV effects were assembled in “transformed parameters” using the independent values sampled from standard normal distribution and OMEGA sampled from Wishart distribution. Six chains, using 500 sampling iterations per chain, were run with each of the parameterizations to evaluate the differences related to estimation time, parameter estimates, and convergence.

The general form of centered and non‐centered parameterizations are shown below for the univariate normal case:

Centered:

$$P_i~\text{lognormal}\left(\log \left(P_{\mathrm{pop}}\right),{\omega }^2\right)$$

Non‐centered:

$$z_i~\text{Normal}\left(0,1\right)$$

$$P_i=P_{\mathrm{pop}}·\exp \left(\omega ·z_i\right)$$

where $P_{\mathrm{pop}}$ and ${\omega }^2$ are sampled population‐level parameters, $P_i$ is the individual‐level parameter. The parameterizations of multivariate–normal–distributed case used in this analysis are provided in Data S6. In the non‐centered parameterization, a Cholesky decomposition was used to account for correlation between the random effects on different parameters (see Data [Link], [Link] for model code of centered and non‐centered parameterizations).

RESULTS

Stan (and the package Torsten) was used to sample the posterior distribution of the PK parameters with 560 observations of 42 participants from Study 004. The parameter estimates with the 90% confidence intervals (CIs) from the initial PopPK model using NONMEM with the FOCEI estimation method ⁴ (Phase II PopPK model and Phase III PopPK model)—the frequentist approach—are provided together with the means and 90% credible intervals (CrIs) from the sampled posterior parameter distributions with the semi‐centered parameterization—the Bayesian approach—in Table 5. Each of the three prior sets had some 90% CrIs that did not cover the previous estimates from the Phase II PopPK analysis. None of the prior sets' 90% credible interval (CrI) covered the previous estimates for Q/F (L/h), residual deviance σ, variance on CL/F, nor covariance on CL/F&Vc/F from the FOCEI estimation.

TABLE 5.

Comparison of estimates/mean sampled value and 90% Confidence Interval/Credible Interval.

	NONMEM FOCEI estimates (Previous analysis)	Uniform prior set				Moderate informative prior set				Very informative prior set
	Estimate (90% CI)	Mean (90% CrI)	Rhat	ESS Bulk	ESS Tail	Mean (90% CrI)	Rhat	ESS Bulk	ESS Tail	Mean (90% CrI)	Rhat	ESS Bulk	ESS Tail
CL/F (L/h)	0.472 (0.406, 0.538) a	0.478 (0.416, 0.545)	1.00	1736.0	1763.7	0.489 (0.429, 0.557)	1.00	2483.0	1976.5	0.524 (0.473, 0.578)	1.00	1924.7	2131.7
Q/F (L/h)	0.014 (0.009, 0.018) a	0.065 (0.039, 0.098)	1.01	609.4	766.8	0.063 (0.038, 0.093)	1.01	937.7	1558.6	0.052 (0.031, 0.079)	1.00	1310.7	1874.1
Vc/F (L)	10.9 (8.696, 13.104) a	6.805 (4.775, 9.503)	1.01	776.9	1240.7	7.563 (5.397, 10.066)	1.00	1036.6	1972.9	9.952 (8.454, 11.431)	1.00	2381.4	1967.5
Vp/R (L)	2.37 (0.314, 4.426) a	2.303 (1.568, 3.125)	1.01	567.2	698.7	2.252 (1.582, 3.077)	1.01	882.1	1609.4	2.042 (1.456, 2.713)	1.00	1163.1	2051.2
Ka (1/h)	0.313 (0.227, 0.399) a	0.178 (0.11, 0.309)	1.01	602.1	957.1	0.194 (0.111, 0.343)	1.01	580.5	1148.1	0.256 (0.139, 0.477)	1.00	812.0	1303.8
Residual Deviance $\sigma$	0.55 (0.475, 0.625) a	0.691 (0.655, 0.728)	1.00	2650.8	2187.2	0.692 (0.657, 0.728)	1.01	3178.4	2449.5	0.694 (0.656, 0.733)	1.00	2005.8	1325.2
Lag time (h)	0.353 (0.184, 0.522) a	1.116 (0.282, 1.625)	1.00	801.9	558.2	1.002 (0.225, 1.589)	1.00	827.7	823.8	0.784 (0.143, 1.436)	1.01	1312.4	1155.1
Weight effect on CL/F and Q/F	1.26 (0.994, 1.526) a	1.258 (0.83, 1.715)	1.00	2139.7	1938.7	1.265 (0.819, 1.703)	1.00	1861.5	1507.2	1.324 (0.892, 1.742)	1.01	1757.6	525.0
Weight effect on Vc/F and Vp/F	1.74 (1.462, 2.018) a	1.341 (0.722, 2.004)	1.00	2240.9	1808	1.354 (0.692, 2.019)	1.00	1810.6	1296.9	1.448 (0.846, 2.067)	1.00	1813.1	1978.7
ADAT effect on CL/F	−0.258 (−0.274, −0.242) b	−0.111 (−0.267, 0.063)	1.00	1335.9	1175.9	−0.115 (−0.268, 0.062)	1.00	1336.3	1406.1	−0.117 (−0.287, 0.054)	1.01	1048.5	986.5
Variance on CL/F	0.247 a	0.058 (0.021, 0.113)	1.03	301.8	740.3	0.060 (0.022, 0.12)	1.01	451.8	1112.2	0.076 (0.026, 0.151)	1.03	286.8	530.3
Covariance on CL/F&Vc/F	0.278 a	0.106 (0.02, 0.224)	1.03	341.6	622.0	0.104 (0.024, 0.223)	1.01	393.3	902.5	0.131 (0.036, 0.264)	1.02	281.5	359.9
Variance on Vc/F	0.343 a	0.352 (0.113, 0.685)	1.02	398.6	817.6	0.320 (0.113, 0.623)	1.01	432.0	769.8	0.325 (0.108, 0.626)	1.03	325.8	355.2
Covariance on CL/F&Ka	–	−0.024 (−0.115, 0.06)	1.01	583.8	847.7	−0.026 (−0.117, 0.059)	1.01	537.9	881.5	−0.017 (−0.133, 0.094)	1.01	603.8	1047.0
Covariance on Vc/F&Ka	–	0.006 (−0.191, 0.235)	1.02	446.7	816.1	0.002 (−0.192, 0.213)	1.01	367.2	655.9	−0.007 (−0.224, 0.245)	1.02	550.4	776.1
Variance on Ka	0.449 a	0.307 (0.069, 0.711)	1.01	336.0	652.3	0.338 (0.07, 0.776)	1.02	310.7	837.6	0.372 (0.077, 0.859)	1.03	205.6	538.0
Covariance on CL/F&ADAT	–	0.005 (−0.04, 0.049)	1.01	656.3	1095.0	0.003 (−0.045, 0.048)	1.00	592.5	956.7	0.002 (−0.06, 0.058)	1.02	348.7	902.0
Covariance on Vc/F&ADAT	–	−0.007 (−0.136, 0.108)	1.01	457.8	827.3	−0.011 (−0.131, 0.11)	1.01	388.6	729.7	−0.013 (−0.145, 0.112)	1.03	286.4	660.4
Covariance on Ka&ADAT	–	0.000 (−0.121, 0.122)	1.01	385.6	771.6	0.003 (−0.134, 0.134)	1.02	228.0	465.9	0.000 (−0.145, 0.145)	1.05	111.8	421.6
Variance of ADAT effect on CL/F	0.030 b	0.096 (0.017, 0.266)	1.02	237.5	516.0	0.095 (0.017, 0.25)	1.01	231.2	387.1	0.098 (0.018, 0.26)	1.03	174.1	385.6

Note: 90% asymptotic confidence intervals were calculated for the fixed effect parameters (THETAs) estimated in NONMEM models. 90% credible interval are shown for all the parameters estimated in Stan models.

Abbreviations: ADAT, Time‐varying Anti‐drug Antibody Status at each observation; CI, confidence interval; CL/F, apparent clearance from central compartment; CrI, credible interval; Ka, absorption rate constant; Q/F, apparent clearance between peripheral and central compartment; Vc/F, apparent central compartment volume; Vp/F, apparent peripheral compartment volume.

^a Estimates using FOCEI method in NONMEM with study CP‐4‐003 (adult) and study CP‐4‐004 (pediatric) (Phase II PopPK model).

^b Estimates using FOCEI method in NONMEM with study CP‐4‐006 (pediatric) (Phase III PopPK model).

The six MCMC chains for each parameter using the three different sets of priors with the semi‐centered parameterization are shown in Figure 1. The weakly informative prior set only provided an upper bound and lower bound for each fixed effect parameter (the priors for IIV parameters were the same in all prior sets), and the chain mixing was similar to the other two prior sets. All three prior sets resulted in acceptable convergence of the chains with all of the $\widehat{R}$ values being less than 1.05 (and most of them were around 1.01).

FIGURE 1.

MCMC trace plot of six chains with 500 warm‐up iterations and 500 sampling iterations for each Stan model. Only the sampling iterations are included in the plot. a: Comparison of trace plot for fixed effect parameters among the three prior sets. b: Comparison of trace plot for random effect parameters among the three prior sets.

In previous Phase II PopPK analysis, the effect of ADAT on CL/F was not estimated due to the limited data. With the same dataset, the posterior distribution of ADAT effect was sampled successfully using each of the three prior sets, though the 90% CrIs of ADAT on CL/F were wide enough to contain 0. Additionally, the previous estimate of ADAT effect in Phase III PopPK analysis (−0.258) was covered by the 90% CrIs using the Phase II study dataset. Although more covariance parameters were included in the presented Bayesian analyses, the sampled posterior distributions of off‐diagonal covariance parameters were centered near zero and their 90% CrIs included zero with the exception of the covariance of CL/F with Vc/F. With each of the three prior sets, the 90% CrIs covered all variance parameters except variance on CL/F, but the ESS (bulk and tail) for IIV parameters were lower than the ESS for other fixed effect parameters using the same prior set.

The posterior predictions of PK concentrations for a representative subset of the pediatric participants are shown in Figure 2. The 90% posterior prediction intervals covered most of the observed concentrations used in the Bayesian model (Study 004 pediatric participants) as well as the observed concentrations of “new” patients (Study 006 pediatric participants, not included in MCMC sampling). Although the sampled posterior distribution for some parameters were slightly different, there is no noticeable difference between the three prior sets in the 90% posterior predicted intervals nor the posterior predicted median values.

FIGURE 2.

Posterior predictions for representative pediatric participants. “Phase II” are the individuals that were included in posterior sampling. “New patients” are the individuals that were not included in posterior sampling (from phase III study).

The MCMC sampling process took 2–3 h to complete on HPC (High‐Performance Computing) server (one core per chain), while the estimation in NONMEM with FOCEI took less than 5 min in the previous analysis (Phase II and Phase III PopPK models). The weakly informative prior set took the longest time, and the very informative prior set took the shortest time. The long running time was mainly because the observations of each individual were across a long period of time. The posterior sampling time with the non‐centered parameterization was faster than the centered parameterization by 20%–30%; our semi‐centered parameterization was about 15% faster than the centered one. With a sufficient number of iterations, the chains from each parameterization mixed similarly well or similarly poorly depending on the prior set. The non‐centered parameterization resulted in generally better ESS than the centered parameterization though the sampled parameter posterior distribution and $\widehat{R}$ were similar to the centered parameterization (Table S1).

DISCUSSION

We implemented Bayesian approaches for a PopPK model using Stan with Torsten on a small dataset of 42 pediatric participants (4–11 years of age) from Study 004 and sampled the posterior predictions for the participants from Study 004 and Study 006 to explore the differences in MCMC sampling among three different sets of priors. Sampled posterior distributions of model parameters were obtained from each prior set and posterior predictions were made for the study individuals as well as for the “new” individuals. The MCMC sampling time was evaluated for both centered and non‐centered parameterizations along with additional optimizations to improve the sampling efficiency.

The prior sets that were defined using information from the previous PopPK analysis ⁴ helped the MCMC chains to sample more reasonable values for each parameter and improved the convergence of each chain. Study 004 was included in the Phase II PopPK analysis, which was the source of the parameter estimates used to define the priors. While this is not an ideal source of “external” information for the Bayesian analysis using Study 004 data, in the presented analysis, only the parameter point estimates were used in moderate and very informative prior sets with considerable large variance values (instead of directly using the estimated variance–covariance matrix). It is recommended to avoid “double‐dipping” approaches when constructing prior distributions. ¹⁹

The sampling time and sampling efficiency was improved when initial values for each of the parameters (with both lower and upper bounds) were provided. Without the initial values, the posterior sampling took a longer time to complete, and the $\widehat{R}$ and ESS were systematically worse, especially for the weakly informative prior set (35% of the 3000 transitions ended up with a divergence, data on file). The lower bound was particularly helpful with the weakly informative prior set as the MCMC chains would occasionally sample into a flip‐flop kinetics solution when the lower bound for Ka was set to 0. Without the lower bound for Ka restricted to be greater than λ₁, the MCMC chains from the weakly informative prior set would not converge for many of the parameters.

It was an interesting finding that the sampled posterior distribution of the parameters did not always align with the parameter estimates from the previous analyses. In many cases, the estimates from Phase II or Phase III PopPK models were outside the 90% CrIs from the informed Bayesian estimates, most notably in the variance parameters (e.g., larger residual variability with smaller IIV on CL/F). It implies that, with the Bayesian approach, some previously identified individual‐level variability is explained through the residual errors. It might also be a result of the limited number of individual participants in the Phase II dataset. While such differences in parameter estimates were expected since the Stan models used a truncated dataset with observations only up to 1 year of dosing (Table 2), it illustrated that the information from priors did not dominate the posterior distribution sampling. The differences in the estimated variance parameters resulted in greater precision than the individual prediction intervals for the participants from the previous PopPK models.

The primary parameter of interest of the Phase III PopPK model, estimated with FOCEI method in NONMEM, was the potential time‐dependent ADA effect (ADAT) on the somatrogon apparent clearance. Since that model treated all the other PK parameters as fixed while the Bayesian approach in this analysis estimated all the parameters, the estimates and CIs/CrIs were slightly different. By accounting for the uncertainty in the PK parameters, rather than fixing them to previous estimates, it is likely that the estimate of the ADAT effect from the Bayesian approach is more accurate. This Bayesian approach may be useful in this case where the effect of a single covariate is of interest and no additional model development or covariate assessment is needed. With all three prior sets, the 90% CrIs of ADAT effect contained zero as well as the estimated value (−0.258). It might be a result of the small sample size of Study 004 that the evidence was not strong enough to support a significant ADA effect.

In addition to exploring the utility of a Bayesian PopPK modeling approach, another goal was to evaluate the performance when predicting PK concentrations for Study 006 participants (Phase III), which were not included in the Stan model (i.e., “new participants”). The 90% prediction interval captured most of the observed concentrations of the representative participants in Study 004 and in Study 006 (new participants), indicating the sampled posterior parameter distributions could explain the trend in the data with the given model and they are general enough for prediction in new participants.

This study shows that embedding the previous knowledge with appropriate variability in the parameter prior could help the MCMC‐sampled chains mix well, and the strength of the prior would not be too overwhelming to the signal in the data. As multiple priors were tested in this analysis (not all results reported), it was noticed that if the prior does not provide enough information (e.g., wide uniform distribution priors or normal distribution prior with large variance), the quality of the MCMC‐sampled chains would completely depend on the signal contained in the data, and it may not fully reveal the advantage of Bayesian inference over other modeling approaches for nonlinear models such as PopPK where the absorption, distribution, and elimination of the drug can have many biologically implausible solutions in the absence of sufficiently dense PK data; on the other hand, if the prior does not provide enough variability (e.g., very narrow bounds for parameters or a normal distribution prior with very small variance), the sampled values of each parameter may be forced to be close to the prior center leading to the information in the data being down weighted in the estimation.

There are many factors that affect the MCMC sampling time in Stan. Having many unknown parameters, complex structure in the model, intensive PK sampling for each participant, collections of PK samples over a long period of time, or many dosing events will increase the sampling time. In PopPK models, when samples are commonly collected over a longer time frame, particularly in later stage studies, more calculations are required resulting in extended sampling time. The simplest way to reduce sampling time is by increasing the computational resources. In our study, the component of the analysis data that most impacted the MCMC sampling time was the number of dosing events. Although the available data of the Study 004 were up to 5.5 years, only samples collected up to one year of treatment for each participant were used to reduce the total estimation time. The non‐centered parameterization also had a meaningful reduction in the run time (by 20–30% for this PopPK model) as well as improving the sampling efficiency. Stan could be more efficient with more optimized parallel processing such as within‐chain parallel computation. ²⁰ , ²¹

Using NONMEM with the $PRIOR subroutine and FOCEI method might be an alternative for the faster‐running time or when the modelers believe the estimates (and the covariance matrix) from a previous analysis are appropriate. Using NONMEM with FOCEI, the model estimation finished in under 5 minutes, while the same model took about 2.5 h on the same HPC grid using Stan. The fully Bayesian approach with Stan does have some clear advantages since the model parameterizations and MCMC sampling are highly flexible and can be applied to a wide range of models, prior distributions, and the posterior distribution from the MCMC process enables more informed—and intuitive—decision‐making and robust inference. ⁸ It is noted that there are Bayesian estimation methods in NONMEM that also can account for parameter uncertainty and they also require longer running time than the FOCEI method.

The study shows an example of fully Bayesian PopPK modeling. Some of the clear benefits from this approach are the intuitive nature of inferences based on posterior distributions, the ability to account for parameter uncertainty in a natural way, and utilize information from proper priors, which makes it a useful tool especially when dealing with small datasets, limited samples, and special populations. Despite the benefits of fully Bayesian modeling, complicated models such as PopPK models present many initial challenges to new users. The syntax to define the Stan model file can be time‐consuming to learn, even for users who are experienced coders. In our experience, the initial Stan model files were very inefficient with the estimation time for this PopPK analysis routinely taking more than 10 h. Without the efficiencies added by using Torsten, the sampling time will be much longer with manually defined ODEs (the Stan model file will also be more complicated). The benefits of a non‐centered parameterization as well as the potential need to declare parameters with more specific bounds that prevent bimodal solutions (e.g., “flip‐flop” kinetics) were also a learning for the authors. Advance optimization techniques, such as within‐chain parallelization, are available to further shorted sampling time, but are not trivial to implement. With these challenges, the benefits of Bayesian approaches to PopPK modeling (and PK/PD modeling) using Stan may not be fully realized, particularly for large datasets where the efficiencies gained from experience are especially impactful. We hope the presented analyses, and the provided code, will be helpful for others considering Bayesian PopPK modeling using Stan and Torsten for their work.

AUTHOR CONTRIBUTIONS

Y.W., X.P., L.F., T.N., and J.K.B wrote the manuscript; Y.W., X.P., L.F., and T.N. designed the research; Y.W and X.P. performed the research and analyzed the data.

FUNDING INFORMATION

The studies were funded by OPKO and this analysis funded by Pfizer Inc.

CONFLICT OF INTEREST STATEMENT

Yuchen Wang is currently an employee of Pfizer and may own stock of Pfizer. Xinyi Pei was an employee of Pfizer at the time of the work. Luke Fostvedt is currently an employee of Pfizer and may own stock of Pfizer. Tao Niu was an employee of Pfizer at the time of the work and may own stock of Pfizer. Joan Korth‐Bradley was an employee of Pfizer at the time of the work and may own stock of Pfizer.

Supporting information

Appendix S1.

Data S1.

Data S2.

Data S3.

Data S4.

Data S5.

Data S6.

Data S7.

Data S8.

Wang Y, Pei X, Niu T, Korth‐Bradley J, Fostvedt L. Implementing a Bayesian approach using Stan with Torsten: Population pharmacokinetics analysis of somatrogon. CPT Pharmacometrics Syst Pharmacol. 2025;14:351‐364. doi: 10.1002/psp4.13279

DATA AVAILABILITY STATEMENT

Upon request, and subject to review, Pfizer will provide the data that support the findings of this study. Subject to certain criteria, conditions and exceptions, Pfizer may also provide access to the related individual de‐identified participant data. See https://www.pfizer.com/science/clinical‐trials/trial‐data‐and‐results for more information.