Mark E. Sale, M.D.: No financial relationships to disclose
Objectives: Assess the performance of multi-objective optimization (MOO) with constraints using NLME and RDarwin for population pk (POPPK) model selection.
Methods: Constraints on model results are common in traditional model selection. The user may require that the final model converge, have a successful covariance step, or describe a quantity (e.g., Cmax, Cmin) with some precision. The single objective model selection solution is to add a large (e.g., 100-point) penalty to the—2ll to drive the model selection toward models that meet this requirement. MOO approaches this problem by adding constraints to model selection, where models that fail the constraint are eliminated from the search. RDarwin [1] is an open-source R package that provides an interface to pyDarwin [2], enabling machine learning model searches from within R. RDarwin provides functions to create the required files to simplify the process of setting up a pyDarwin search. A simulated data set with 16 subjects and 112 observations was created. Data were simulated from an IV bolus 1-compartment model with a single dose of 25000 and 8 samples each at random times over 24 hours. The NSGAIII algorithm[3], with the pymoo python package [4] was used. Search configuration: Structural: 1, 2, or 3 compartments Error models: proportional; combined additive + proportional Covariate relationships: CL and Vc as functions of age, weight, and/or sex Random effects: with or without BSV on intercompartmental clearances and peripheral volumes Three objectives were included for the MOO model selection: −2 log likelihood (−2LL) Model complexity (Number of estimated parameters) Cmax fit penalty: 300*(abs(Cmax(observed) – Cmax(predicted)/Cmax(observed)))
The NSGA III algorithm (pymoo) was executed via RDarwin with 6 and 10 partitions. Partitions are used to ensure diversity in the final non-dominated set. Searches were run with and without the constraint RSE < 0.5 on all parameters, eliminating any infeasible solutions.
Results: The constrained search returned 7 non-dominated models for both the 6 and 10 partition searches. All non-dominated models had all RSE values < 0.5. The unconstrained search returned 10 models for 6 partitions and 11 for 10 partitions. Only 1 model for the unconstrained with 6 partitions and 2 models for the unconstrained 10 partitions search had all RSE’s < 0.5. Of the 3 models in common between the unconstrained and constrained searches, all were identical. Below is a table of the 3 objectives of the models returned by the 6 partition MOO constrained search -2ll NParms Cmax Maximum Penalty RSE 1,223.55 14 24.17 0.45 1,269.41 8 30.33 0.37 1,261.02 13 23.80 0.48 1,260.94 10 23.92 0.48 1,269.55 7 30.3 0.37 1,232.69 9 30.78 0.46 1,229.75 11 25.03 0.48
Conclusions: Constrained MOO reliably identifies a concise set of Pareto optimal PopPK models that all satisfy convergence, covariance, and precision (RSE < 0.5) requirements, without loss of diversity compared to unconstrained searches. These findings endorse constraint based MOO as a rigorous, transparent approach for pharmacometric model selection in both simulated and clinical settings.
Citations: [1] https://certara.github.io/R-Darwin/ Accessed 19 April 2025 [2] https://certara.github.io/pyDarwin/html/index.html. Accessed 19 April 2025 [3] https://pymoo.org/algorithms/moo/nsga3.html Accessed 19 April 2025 [4] https://pymoo.org/ Accessed 19 April 2025
Keywords: Machine learning, population pharmacokinetics
