Development and exploration of exhaustive, stepwise, and heuristic algorithms for automated population pharmacokinetic modelling
Zhonghui Huang(1), Joseph F Standing (1), Frank Kloprogge (2)
(1) Infection, Immunity and Inflammation Research & Teaching Department, UCL Great Ormond Street Institute of Child Health, London, United Kingdom (2) UCL Institute for Global Health, University College London, London, United Kingdom
Objectives: Automated population pharmacokinetic modelling tools are currently being investigated. The genetic algorithm (GA) has been proposed to address pharmacometric model selection in NONMEM  and MONOLIX . Overfitting of automated model building algorithms may be a problem and has not been fully addressed.
Furthermore, other methods are available, one such being the ant colony optimisation (ACO) algorithm inspired by ant foraging behaviour in the nature . ACOs have been compared with GAs and found to be more efficient and accurate in other fields .
The aim of this study was to compare the model performance identified by stepwise algorithm, GA and ACO with the exhaustive approach providing ground truth.
Methods: All algorithms for automated modelling were developed in R software  and modelling work was undertaken in nlmixr2  using SAEM algorithm.
Simulated datasets. A total of 72 single-dose intravenous simulated datasets were generated, with 24 cases of sparse sampling, 24 cases of rich sampling, and 24 cases of optimal sampling design. Each set was generated from six one-, nine two- and nine three-compartment models. PK parameter values were informed by a trend analysis . Model datasets were intravenous cases with the default dose set as 100 mg. Sampling design was derived by ED-design and subject size ranged from 10 to 150.
Search space. The search space consisted of 243 structural and statistical models including: 1, 2 or 3 compartment, varying inter-individual variability, eta-correlation, and additive, proportional or mixed residual error models.
Fitness function. Candidate fitness functions tested were information criterion consisting of Akaike information criterion (AIC), Bayesian Information Criterion (BIC) and Objective function values (OFV) without and with penalty points ranging from 10 to 10,000 on RV variance, RV variance and relative standard errors (RSE%) and RV variance, RSE% and shrinkage.
Stepwise algorithm. The base model of this stepwise approach was one compartment only with random effect on clearance, and additive plus proportional residual error. The models were explored in a stepwise manner, namely, structural, inter-individual variability and residual error model.
GA and ACO. Both GA and ACO were allowed to select from the 243 models. GA incorporated key design elements that have been retained from previous studies [1,2]. Tournament selection combined with an elitism strategy was used, and for every 3 generations, a local exhaustive search was conducted. The population size was set as 10 with 0.2 mutation rate. ACO was designed based on a simple ACO (S-ACO) algorithm . Key parameters were the number of ants for each travel (10), the rate of pheromone evaporation (0.2) and the initial pheromone (1). Pheromone generation was calculated as the inverse of the historical fitness function ranking. An elitism strategy was also used.
The “true best models” determined by the exhaustive search were compared with the results of each algorithm. The accuracy rate was defined as the percent agreement between developed algorithms and the exhaustive search.
Results: AIC or OFV with penalty set to a constant of 10,000 on RSE, shrinkage and RV variance outperformed all other fitness functions. These criteria recovered the structural and RV model in 96% and 100% of optimal design cases. Inter-individual variability was correctly recovered in 74% of cases for successfully identified structural models.
Poor performance, an overall accuracy rate of 46%, for identification of “true best model” by stepwise algorithm was observed with 24 optimal design datasets. In contrast, GA and ACO achieved higher accuracy (76% and 80%).
For rich data, a similar trend was observed where stepwise algorithm displayed poor performance with an accuracy rate of 50% while GA and ACO achieved accuracy rates of 69% and 68%, respectively. However, the stepwise algorithm and ACO were able to identify the "true best model" in 79% and 86% of cases with sparse data whilst the GA only achieved this in 69% of cases.
Conclusions: The ACO outperformed the GA in general using a smaller number of models while the stepwise method did not reliably find the true best PopPK model in the search space. ACO, therefore, has the potential to reduce labour and the computational burden. This study focused on intravenous bolus administration only and further investigation for other routes of administration is required.
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