The drug titration paradox in the presence of intra-individual variation: can we estimate the true concentration-effect relationship?
Sebastiaan C. Goulooze (1), Elke H.J. Krekels (2,3) Martijn van Noort (1), Catherijne A.J. Knibbe (3,4)
(1) LAP&P Consultants BV, Leiden, The Netherlands (2) Certara Inc, Princeton, NJ, USA (3) Division of Systems Pharmacology and Pharmacy, Leiden Academic Centre for Drug Research, Leiden University, Leiden, the Netherlands (4)Department of Clinical Pharmacy, St. Antonius Hospital, Nieuwegein, The Netherlands .
Objectives: The drug titration paradox arises when higher drug concentrations are paradoxically associated with poorer efficacy outcomes, due to the titration of the drug dose to achieve a desired effect [1]. In situations where there is inter-individual variability in the pharmacodynamic outcome, the sickest patients receive higher doses, resulting in elevated drug concentrations. Consequently, at the population level, the observed correlation between drug concentration and outcome may not accurately represent the true relationship between drug concentration and its causal effect on outcome [1].
If the drug titration paradox occurs at the population level (and not at the individual level) unbiased estimates of the concentration-effect relationship can be obtained using non-linear mixed-effects models [2]. However, the drug titration paradox may be observed on the individual level when dose titration occurs in a setting with considerable intra-individual variability of disease severity [3]. In such a scenario, it remains unknown whether unbiased estimates of the concentration-effect relationship can be obtained, and which methodology would be most appropriate.
Using morphine treatment of pain as an example, this study assesses estimated parameter bias by simulating a morphine titration study with substantial intra-individual variability and re-estimating the concentration-effect relationship with different analysis approaches.
Methods: For 100 patients in 100 datasets, pain scores ranging from 0 to 10 were simulated every 2 hours over 48 hours using NONMEM 7.5.1. Each patient received a single 5 mg morphine loading dose at time=0. A titration algorithm guided subsequent morphine doses: no additional morphine for pain scores <3, bolus of 5 mg for pain scores of 3-6, and 10 mg for scores above 6. A published population pharmacokinetic (PK) model for morphine was used to simulate morphine concentrations over time based on individual doses [4]. The morphine concentration-pain relationship was characterized by a direct effect Emax model with interindividual variability on baseline pain, EC50, and Emax. Intra-individual variability in disease severity (i.e., baseline pain) was simulated as Brownian motion.
Three model fitting approaches were applied in NONMEM 7.5.1 to each dataset: The true PKPD model using stochastic differential equations (SDE) with an Extended Kalman Filter to characterize intra-individual variations of the baseline pain [5]. Additionally, two misspecified models were tested, one in which SDEs were omitted and another in which SDEs were replaced with inter-occasion variability (IOV), with each occasion lasting 12 hours. Parameter estimates were compared to the true values to calculate the bias for each simulated dataset.
Results: Analysis of the first pain scores, which were obtained before the first dose titration, revealed a correlation between morphine concentration and pain scores consistent with the causal drug effect. For later timepoints, a reversal of this trend was observed; due to the negative feedback loop of the titration algorithm, higher morphine concentrations were associated with higher pain scores.
Estimating with the true model (i.e., including SDE) resulted in median biases of 8.3% and 2.6% for PD parameters EC50 and EMAX, respectively. The misspecified models resulted in biased parameter estimates, especially for Emax. The model with SDE omitted had substantial bias in the Emax, with estimates often close to 0 (median bias -101.9%), resulting in a reversed direction of the drug effect in 56/100 datasets. The addition of IOV to this model mitigated this to some extent, but substantial bias persisted (median bias Emax -53.3%).
Conclusions: Our findings emphasize that, even in the presence of strong intra-individual variability and drug titration, an unbiased population estimate of the concentration-effect relationship can be obtained with a model that characterizes intra-individual variability of disease severity, such as the model with SDE. This study underscores the critical importance of appropriately characterizing intra-individual variability in titration studies, as the misspecified models yielded significantly biased estimates of the drug effect and even occasionally a reversal of the direction of the drug effect (i.e., higher drug concentrations are associated with more pain). Such biased estimates of drug effects could lead to incorrectly concluding a lack of efficacy for a promising drugs.
References:
[1] Schnider et al. Clin Pharmacol Ther (2021) 110:401-408.
[2] Kristensen et al. CPT: Pharmacometr Syst Pharmacol (2022) 11:1592-1603.
[3] Schnider et al. Br J Anaesth (2022) 129:861-867.
[4] Ahlers et al. Anesth Analges (2015) 121:1261-1273.
[5] Tornøe et al. Pharmaceut Res (2005) 22:1247-1258.
