Derivation and theoretical validation of fractional quasi-steady state approximation (fQSSA) for target-mediated drug disposition models with memory effects

Abstract

Standard target-mediated drug disposition (TMDD) models are widely used to describe nonlinear pharmacokinetics driven by high-affinity drug-target interactions. However, their reliance on instantaneous binding limits their ability to capture delayed and history-dependent dynamics observed in vivo. Here, we introduce a fractional TMDD model that incorporates memory effects through a fractional derivative, thereby generalizing the standard TMDD (sTMDD) framework. Although this fractional TMDD (fTMDD) formulation increases modeling flexibility, it also exacerbates parameter identifiability challenges under typical experimental conditions where only drug concentration data are available. To address this limitation, we derive a fractional quasi-steady-state approximation (fQSSA) that reduces model dimensionality while preserving essential nonlinear and memory-dependent pharmacokinetic dynamics. We further establish an explicit validity condition that quantifies the approximation error of both fTMDD and fQSSA without requiring numerical simulation. This condition reveals that the initial drug-to-target ratio is the primary determinant of QSSA validity, whereas the fractional order has a comparatively minor influence. Application of the proposed framework to recombinant human erythropoietin (rhEPO) data demonstrates that fractional dynamics play a population-dependent role, improving model performance in adults but not in infants. Together, this work provides the first systematic derivation of a QSSA framework for fractional TMDD models, along with rigorous and computable applicability conditions. Our results establish a principled foundation for incorporating memory effects into pharmacokinetic modeling and offer a generalizable framework for nonlinear PK-PD systems involving binding-mediated dynamics. Author summaryMany drugs interact strongly with their biological targets, leading to complex and nonlinear pharmacokinetics that are commonly described using target-mediated drug disposition (TMDD) models. However, these models assume that drug-target interactions occur instantaneously, which limits their ability to capture delayed and history-dependent behaviors observed in real biological systems. In this study, we develop a new modeling framework that incorporates such memory effects by extending TMDD models using fractional calculus. To make the model more practical and computationally efficient, we derive a simplified version based on a quasi-steady-state approximation (QSSA) and provide a clear mathematical condition that determines when this simplification is valid. Our analysis shows that the accuracy of the simplified model is primarily controlled by the initial ratio of drug to target, while the influence of memory effects is comparatively smaller. When applied to experimental data for erythropoietin, our model reveals that memory effects are important in adults but negligible in infants, suggesting that these effects may reflect underlying physiological differences. Overall, this work provides a systematic and interpretable framework for incorporating memory effects into pharmacokinetic modeling, with potential applications to a wide range of drug systems involving complex binding dynamics.