A. Janssen1, F.C. Bennis2,3, M.H. Cnossen4, and R.A.A. Mathot1
[1] Department of Hospital Pharmacy & Clinical Pharmacology, Amsterdam UMC, University of Amsterdam, The Netherlands. [2] Emma Children’s Hospital, Amsterdam UMC, University of Amsterdam, Follow Me & Emma Neuroscience Group, Meibergdreef 9, Amsterdam, The Netherlands. [3] Amsterdam Reproduction and Development, Amsterdam, The Netherlands. [4] Department of Pediatric Hematology, Erasmus MC Sophia Children’s Hospital, Erasmus University Medical Center Rotterdam, The Netherlands.
Introduction In mixed-effect models, random effects are used to explain residual inter-individual variability. Their effect is constant over time, and is generally unsuited to represent time-dependent changes in model parameters. Disregarding such changes can lead to significant model error in cases where disease state or organ function is highly variable (e.g. renal/hepatic function impacting drug clearance). One approach to handle this data is to discretize the time-window in order to re-estimate random effects at certain time-points (e.g. to describe inter-occasion variability; IOV). Selecting the appropriate time-points at which to make such adjustments a priori is difficult, while frequent re-estimation over small time frames risks overfitting.
In this work, we suggest using Gaussian Processes (GPs) to represent time-dependent random effects. GPs are an infinite-dimensional extension of the multivariate normal distribution, and can be thought of as representing IOV in continuous-time or describing distributions over functions[1]. Prior distributions over these functions can be learned, allowing for the determination of typical effects as well as the degree of variability between subjects.
Methods We test the approach by simulating data for 60 subjects receiving a hypothetical drug following a one-compartment model with first-order elimination from the central compartment (k10 = 0.36) and first-order absorption (ka = 0.1). Covariate effects included allometric scaling of clearance and volume of distribution, and higher clearance for female patients (+22%). A sinusoidal time-dependent effect f(t) = α⋅sin(2πt/96) + 1 was added to k10 with varying amplitude α ~ Beta(4,5) for each subject. Each subject received a daily dose of 0.3 mg/kg for five subsequent days, and three measurements were collected every 1.5, 12, and 24 hours after administration. Additive error (σ = 0.5 mg/L) was added to the measurements. Model parameters were optimized using a Variational Expectation Maximization procedure, first fitting individual posterior distributions over the random effects followed by the estimation of fixed-effects parameters and GP hyper-parameters[2,3].
We also evaluate the model on a real-world data set of 47 haemophilia A patients undergoing surgery[4]. Here, factor VIII clearance might be variable over time[5,6]. Median follow-up time was 190.5 hours and daily peak and trough FVIII levels were collected (total of 455 measurements). We extend a population PK model developed on this data set with a time-dependent random effect on clearance to identify patients with relevant effects (>15% maximal change in clearance).
Results In the simulation experiment, the addition of time-dependent effects improved prediction accuracy (root mean squared error = 0.41 mg/L, R2 = 0.94 compared to 1.06 mg/L, R2 = 0.62 without time-dependent effects). Similarly, estimated model parameters were highly accurate: typical k10 estimate was 0.39 (true = 0.36), ka was 0.10 (true = 0.10), change in clearance when female was +20% (true = +22%), and additive error estimate was 0.41 (true = 0.50). The model was able to learn varying time-dependent effects on an individual basis, and captured the sinusoidal pattern.
On the real-world data set, we found that roughly half of subjects (n = 25) depicted relevant time-dependent effects. Effects were highly variable between patients but there seemed to be a general trend of decreased FVIII clearance in the first three days after surgery. We found that older patients and those undergoing medium risk surgical procedures longer than two hours had significant higher likelihood (p < 0.05; logistic regression) of presenting relevant time-dependent effects.
Conclusion We propose the use of GPs to model time-dependent random effects. GPs allow for the learning of prior distributions over their effects, constraining the solution to follow learned patterns (reducing overfitting). Based on the perioperative data, we found that time-dependent effects could be highly variable between subjects, indicating the challenge in estimating such effects using deterministic functions or typical effects. The GP posterior can be iteratively updated as more data becomes available, and the effect can be projected into the future while taking into account uncertainty over its trajectory. The approach is thus well suited for the real-time optimization of treatment in complex clinical settings.
References:
[1] Rasmussen CE, Williams CKI. Gaussian Processes for Machine Learning. The MIT Press; 2005.
[2] Titsias M. Variational Learning of Inducing Variables in Sparse Gaussian Processes. Proceedings of the Twelth International Conference on Artificial Intelligence and Statistics. 2009;567–574.
[3] Quinonero-Candela, Joaquin, and Carl Edward Rasmussen. “A unifying view of sparse approximate Gaussian process regression.” The Journal of Machine Learning Research 6 (2005): 1939-1959.
[4] van Moort, I., Preijers, T., Bukkems, L. H., Hazendonk, H. C., van der Bom, J. G., Laros-van Gorkom, B. A., … & Keeling, D. (2021). Perioperative pharmacokinetic-guided factor VIII concentrate dosing in haemophilia (OPTI-CLOT trial): an open-label, multicentre, randomised, controlled trial. The Lancet Haematology, 8(7), e492-e502.
[5] Longo G, Messori A, Morfini M, et al. Evaluation of factor VIII pharmacokinetics in hemophilia-A subjects undergoing surgery and description of a nomogram for dosing calculations. Am J Hematol. 1989;30(3):140–149.
[6] Batorova A, Martinowitz U. Intermittent injections vs. continuous infusion of Factor VIII in haemophilia patients undergoing major surgery. British Journal of Haematology. 2000;110(3):715–720.
Reference: PAGE 32 (2024) Abstr 10939 [www.page-meeting.org/?abstract=10939]
Poster: Methodology - New Modelling Approaches