Exact Formula of the Total Quasi-Steady State Approximation in Competitive Target-Mediated Drug Disposition

Competitive target-mediated drug disposition (competitive TMDD) arises when 2 drugs compete for the same target receptor. The full competitive TMDD model characterizes these dynamics; 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, which remains valid for broad parameter regimes, the exact formula of the total quasi-steady state approximation-based reduced model-competitive qTMDD-remained unknown for 15 years, as it requires solving a cubic equation and identifying a biologically meaningful solution. Consequently, researchers have relied on computationally expensive numerical approximations, limiting practical applicability. Here, 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 formulation improved computational speed by approximately 18-fold compared to numerical approximation methods, enabling competitive qTMDD-based Bayesian inference that was previously impractical. Applied to clinical trial data for anakinra and rhIL-7-hyFc, competitive qTMDD estimated pharmacological parameters comparable to those from the full competitive TMDD model while requiring only approximately 28% of the computation time. Importantly, it maintains robust accuracy even at low sampling density, whereas previous reduced models exhibit bias. By ensuring precise biological interpretation of drug systems even in complex real-world scenarios, the exact formula of competitive qTMDD has the potential to 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.