The Neural Mixed Effects algorithm: leveraging machine learning for pharmacokinetic modelling
A. Janssen(1), F.W.G. Leebeek(2), M.H. Cnossen(3), and R.A.A Mathôt(1) for the OPTI-CLOT study group and SYMPHONY consortium
(1) Department of Hospital Pharmacy - Clinical Pharmacology, Amsterdam University Medical Center, Amsterdam, The Netherlands. (2) Department of Hematology, Erasmus MC, Erasmus University Medical Center Rotterdam, The Netherlands. (3) Department of Pediatric Hematology, Erasmus MC Sophia Children’s Hospital, Erasmus University Medical Center Rotterdam, The Netherlands.
Objectives: There is much interest in the application of machine learning (ML) algorithms for the personalisation of treatment. Nonlinear mixed-effects (NLME) models are pivotal for current therapeutic drug monitoring strategies but may also have limitations. One such limitation is in handling particularly high dimensional and complex data. For genomics data for example, interactions between a multitude of genes are difficult to represent in an algebraic form. Instead, ML algorithms can learn these relationships directly from data, providing a solution to this issue. Unfortunately, ML algorithms are difficult to apply due to the following three issues:
- Most datasets available to pharmacologists are small, with sparse drug concentration measurements. This makes it difficult for ML algorithms to learn the relationship between the covariates and the measurements.
- Patient dosing schemes can be very complex. Most ML algorithms do not allow for a simple way of implementing dosing events over time.
- ML algorithms do not consider residual inter-individual variability (IIV). In contrast, NLME models quantify several levels of variability which potentially represent the effect of unknown covariates or measurement noise. If unaccounted for, the model might incorporate noise in its predictions.
To resolve the abovementioned issues, we present the neural mixed effects (NME) algorithm. This algorithm incorporates a neural network (NN) into the mixed effects statistical framework. We will present a dataset of simulated haemophilia A patients receiving blood clotting factor VIII (FVIII) to test the accuracy of the NME algorithm.
Methods: The NME algorithm was developed in the Julia programming language (Julia Computing, Inc., v1.6.0). The basis of the NME algorithm is a NN that predicts the parameters for a system of ordinary differential equations (ODEs) representing a compartment model. This allows us to pass dosing information directly to the ODE solver. Next, we use the first order approximation of the extended least squares objective function [1]. Using this objective function we can simultaneously update the weights of the NN as well as the population parameters representing IIV. The resulting NME algorithm represents the best of two worlds: it provides a measure of IIV allowing individualized predictions like in NLME models, while using the function approximation capabilities of NNs for the implementation of covariates.
We simulated 500 patients based on the NONMEM model by Hazendonk et al [2]. This model was developed using data from 119 haemophilia A patients treated with FVIII concentrate perioperatively. IIV was implemented on clearance (CL, ω2=0.129) and central volume of distribution (V1, ω2=0.0705). Each individual received a 1500 IU dose at t=0. FVIII plasma levels were simulated based on individually predicted PK parameters at t=0.08, 0.5, and every hour until t=48. Training samples were limited to t=0.5, 4, 12, and 48 hours. Average simulated FVIII peak level directly after dose was 0.630 IUmL-1 [0.215-4.30]. Random noise (σ=0.04) was added to the simulated FVIII levels. From the dataset, 75 randomly selected patients were used to train the NME model. The NME model was based on a two-compartment model, with IIV on CL and V1. An additive residual error model was used. Used covariates were weight, age, blood group and estimated surgical intensity. Model accuracy was represented as the percentage of predictions within 0.05 IUmL-1 of the true plasma level (without noise) for all 50 timepoints of the remaining 425 patients.
Results: On the validation set, 68.5% of typical FVIII predictions by the NME model were within 0.05 IU mL-1 of the true concentration. For the individual predictions, 92.0% were within this range. The NME model accurately predicted IIV parameters for CL (ω2 = 0.126 vs 0.129) and V1 (ω2 = 0.0677 vs 0.0705). In addition, the model predicted variance of residual error was 0.0385, close to the true value 0.04 used in the simulation.
Conclusion: Our results show that the NME algorithm accurately predicts FVIII levels and IIV based on limited drug measurements. In contrast to other ML algorithms [3,4], the NME algorithm supports any dosing scheme and incorporates prior knowledge about drug dynamics through a compartment model.
In conclusion, our results indicate that by combining NLME models and ML, the NME algorithm can prove to be a valuable tool for performing PK analyses.
References:
[1] Beal, S.L., Sheiner, L.B., Boeckmann, A.J. & Bauer, R.J. (Eds). NONMEM VI user guides ( ICON plc, Gaithersburg, MD, 1989–2021). <https://nonmem.iconplc.com/nonmemVI/guides>.
[2] Hendrika Hazendonk, Karin Fijnvandraat, Janske Lock, Mariëtte Driessens, Felix van der Meer, Karina Meijer, Marieke Kruip, Britta Laros-van Gorkom, Marjolein Peters, Saskia de Wildt, Frank Leebeek, Marjon Cnossen, Ron Mathôt. A population pharmacokinetic model for perioperative dosing of factor VIII in hemophilia A patients. Haematologica 2016;101(10):1159-1169; https://doi.org/10.3324/haematol.2015.136275.
[3] Brier, M. E., Zurada, J. M., & Aronoff, G. R. (1995). Neural network predicted peak and trough gentamicin concentrations. Pharmaceutical research, 12 (3), 406-412.
[4] Liu, R., Li, X., Zhang, W., & Zhou, H. H. (2015). Comparison of nine statistical model based warfarin pharmacogenetic dosing algorithms using the racially diverse international warfarin pharmacogenetic consortium cohort database. PloS one, 10 (8), e0135784.
