Exact formula of the total quasi-steady state approximation in competitive target-mediated drug disposition

Abstract Competitive target-mediated drug disposition (competitive TMDD) arises when two drugs compete for the same target receptor. These dynamics can be characterized by the full competitive TMDD model; yet, its complexity motivated the use of a reduced model, which is invalid under high receptor concentrations. While this problem can be resolved by using the total quasi-steady state approximation (tQSSA), which remains valid for all receptor conditions, the exact formula of the tQSSA-based reduced model—competitive qTMDD—has remained unknown for 15 years, as it requires solving a cubic equation and the nontrivial task of identifying a biologically meaningful solution. Consequently, researchers have relied on numerical approximation methods, which impose substantial computational costs to maintain accuracy and thereby hinder their practical use in complex real-world applications. To address this problem, we derive—for the first time—a real-valued exact formula for competitive qTMDD by leveraging analytic properties of cubic equations and geometric characteristics of their roots in the complex plane. This exact formula improved computational speed by more than 11-fold compared to previous numerical approximation methods, thereby enabling Bayesian inference using competitive qTMDD, which had been impractical due to excessive computational time. When applied to real-world data from clinical trials, competitive qTMDD estimated pharmacological parameter estimates comparable to those from the full competitive TMDD model while requiring only 30-43% of the computation time. Importantly, this estimation using competitive qTMDD remained consistently accurate regardless of data sparsity, whereas the previous reduced model produced biased estimates under sparse sampling conditions. By ensuring precise biological interpretation of drug systems even in complex real-world scenarios, the exact formula of competitive qTMDD has the potential to significantly streamline the drug development and clinical testing process. Our exact formula also consists entirely of real-valued terms, allowing seamless integration into existing pharmacometrics software. Author Summary Competitive target-mediated drug disposition (competitive TMDD) occurs when two drugs compete for the same target receptor. Analyzing this interaction has faced a dilemma for 15 years: choosing between a ‘full model’ that is accurate but computationally intensive, and a ‘reduced model’ that is fast but often loses accuracy under real-world clinical conditions. Researchers tried to solve this dilemma by deriving an accurate reduced model; however, it was mathematically challenging. As a result, they had to rely on numerical approximations—which require substantial computational power to maintain the accuracy needed for clinical use, making them impractical in real-world scenarios. In this study, we derive a first-ever exact formula for the new reduced model that is accurate across all biological conditions. This formula computes more than 11 times faster than previous numerical methods, making advanced statistical analyses—such as Bayesian inference—feasible for the first time in this context. When applied to real-world clinical data for Anakinra and rhIL-7-hyFc, our method yielded parameter estimates as accurate as the full model but required significantly less computation time. This breakthrough provides a more accurate biological interpretation and better guidance for determining the right drug dose, potentially accelerating drug development and reducing associated costs.