Exposure–response modelling approaches for determining optimal dosing rules in children

Age is more than a number: Exposure–response modelling approaches for determining optimal dosing rules in children

PHASTAR statistician Ian Wadsworth has had a new research article published in Statistical Methods in Medical Research examining modelling approaches for quantifying how exposure-response parameters vary over different ages in paediatric populations. Here he summarises the methodologies considered: 

Within paediatric populations, different age groups of children treated with a new medicine may experience differences in dose–exposure and exposure–response (E–R) relationships due to age-related differences in growth, development and physiological differences [1]. One possible suggested age grouping by the International Council for Harmonisation (ICH) E11 document is preterm newborn infants; term newborn infants (0–27 days); infants and toddlers (28 days to 23 months); children (2–11 years); and adolescents (12 to 16–18 years, depending on region) [1]. However, such groupings are general suggestions, with specific treatments and therapeutic areas having the potential for differences. Our paper [2] developed approaches to estimate the E–R relationship in children and to identify age groupings which define practical and effective dosing rules.

Consider the simple scenario where a linear model is assumed for the relationship between exposure and response. For example, the relationship between response of log-transformed percent change from baseline in seizure frequency and the average steady-state trough concentration for the treatment of partial onset seizures with topiramate [3]. However, more complex models can also be adopted. We consider several model-based approaches to quantify how E-R model parameters vary over age: model-based recursive partitioning (MOB), a variation on this called partially additive linear model (PALM) trees, and Bayesian penalised B-splines. 

MOB allows data to be split into groups based on partitioning variables, with each subgroup characterised by its own parametric model [4]. We consider a MOB approach using age as the only partitioning variable and fitting a linear model within each partitioned group. The MOB algorithm can be implemented using the ‘mob’ function found in the R ‘partykit’ package [5] and can be extended beyond our linear model example to any parametric model. PALM trees are a variation of MOB which allow for additional global parameters which remain constant across the partitioning groups. However, where MOB allows for any parametric model, PALM trees restrict to generalised linear models. The PALM tree algorithm can be implemented using the ‘palmtree’ function found in the R ‘partykit’ package [6]. 

The MOB and PALM trees procedures output a tree where each split gives a subgroup with a fitted parametric model. We bootstrap the E-R data, fit a tree to each sample and from each tree estimate age-specific intercept and slope at values on a range from 0 to 18 years. For each age point, we aggregate across the bootstrap samples and obtain an estimate of the E–R intercept or slope for any given age by linear interpolation. This allows us to have an estimate of how the intercept and slope parameters vary over. 

Below we present an example of this PALM tree procedure being applied to simulated E-R data. The data has been simulated such that there are 100 subjects in total, split into four equally sized age groups with distinct linear E-R relationships (0-4 years, 4-10 years, 10-14 years, 14-18 years), with corresponding true intercepts (5.1, 4.8, 4.4, 3.9) and slopes (0.01, 0.035, 0.075, 0.125) in each group. The below plot shows how the estimate of intercept changes with age:

Our next approach is the use of Bayesian penalised B-splines [7]. Splines define flexible regression models by joining smooth curves in subintervals together at knot points and model how the intercept and slope of our linear model varies with age. As before, we present a plot of how the estimate of intercept changes with age from an example of this Bayesian penalised B-splines procedure being applied to the same simulated E-R data:

From this plot, we see how the B-splines are more flexible, adapting more to the data whilst still capturing the overall change in intercept over age. 

After estimating how exposure–response model intercept and slope parameters vary with age, we propose an approach for deriving optimal dosing rules across these age groups. When defining the target exposure for each age group, we would like to minimise the difference between the expected response and a target response. One approach would be to find the dosing rule minimising the total absolute difference between the expected and target response. As we are working in the setting where we believe E-R model parameters may depend on age, it is plausible that parameters may change rapidly over narrower age ranges (such as young children aged 0 to 2 years) compared to wider age ranges, such as adolescents. As such, we choose to minimise the absolute difference between expected and target response incorporating weights within pre-specified age groups, for example based on the ICH age groupings mentioned earlier. This weights by the width of each pre-specified age range, allowing narrower age groups to contribute to the optimisation to a greater extent. 

We performed a simulation study to evaluate how well our approaches estimate the intercept and slope parameters and the accuracy of recommended dosing rules. We considered a range of scenarios varying the underlying age groups in terms of number, range and linear model parameters. We measured how well each method estimated the true intercept and slope parameters by calculating average absolute bias, empirical standard deviation and empirical mean squared error of estimates over 1000 simulated datasets at each age in a grid over 0 to 18 years. Numerical integration was then performed over the 0 to 18 years age range to get overall measures of accuracy, precision and mean squared error for how each method performed in estimating the intercept and slope parameters across scenarios. 

Results suggested that across the scenarios we considered, both PALM trees and Bayesian penalised B-splines can estimate underlying changes in linear E-R model parameters well, with the Bayesian penalised B-splines approach consistently estimating the intercept and slope more accurately than the PALM trees. 

If you’d like to read more details on the models outlined and an application to an in vitro study of cyclosporine, you can access Ian’s paper here: https://journals.sagepub.com/doi/10.1177/0962280220903751


References

  • International Council for Harmonisation of Technical Requirements for Pharmaceuticals for Human Use (ICH). E11: Note for guidance on clinical investigation of medicinal products in the paediatric population. CPMP/ICH/2711/99.
  • Wadsworth I, Hampson LV, Bornkamp B, and Jaki T. (2020). Exposure–response modelling approaches for determining optimal dosing rules in children. Statistical Methods in Medical Research. https://doi.org/10.1177/0962280220903751
  • Girgis IG et al. (2010). Pharmacokinetic-pharmacodynamic assessment of topiramate dosing regimens for children with epilepsy 2 to <10 years of age. Epilepsia, 51(10):1954-62
  • Zeileis A, Hothorn T and Hornik K. (2008) Model-based recursive partitioning. Journal of Computational and Graphical Statistics, 17: 492–514.
  • Hothorn T and Zeileis A. (2015) Partykit: a modular toolkit for recursive partytioning in R. Journal of Machine Learning Research, 16:3905–3909.
  • Seibold H, Hothorn T and Zeileis A. (2019) Generalised linear model trees with global additive effects. Advances in Data Analysis and Classification. 13: 703–725.
  • Eilers PH and Marx BD. (1996) Flexible smoothing with B-splines and penalties. Statistical Science, 11: 89–102.