Asymptotic Analysis on a TMDD model: Control of the process
Dimitris T. Maris (1), Dimitris G. Patsatzis (1) and Dimitris A. Goussis (1).
Department of Mechanics, School of Applied Mathematics and Physical Sciences, National Technical University of Athens, 157 80 Athens, Greece
Objectives: A detailed analysis of a multi-scale pharmacokinetic-pharmacodynamic (PKPD), one compartment, target mediated drug disposition model (TMDD) is performed. This TMDD model incorporates the interaction of a drug with its target, the binding of the compounds (generation of the complex) and the outcome of their interaction. The purpose of this analysis is to identify methodologies for the control of the process by acquiring a full system-level understanding.
Methods: The analysis is based on the Computational Singular Perturbation (CSP) algorithm [1]. CSP provides (i) an approximation of the Slow Invariant Manifold (SIM), which is the surface created by the constraints of the system (the emerging equilibria), (ii) the reduced model, that drives the system along the SIM and (iii) a number of diagnostic tools, that can be employed for the identification of the reactions that are responsible for the formation of the SIM and the reduced model. Among others, this method can identify numerically the stages in the evolution of the process where Quasi Steady State (QSS) or Partial Equilibrium (PE) approximations are valid [2].
Results: The reactions in the model that (i) generate the fast time scales, (ii) contribute to the formation of the SIM and (iii) drive the slow system, were identified. The analysis concluded that there are two distinct stages in the evolution of the process where QSSA and PEA are valid. The parameters of the model that (i) affect the formation of the SIM and (ii) the way the solution lands and then evolves on it were identified.
Conclusions: The present analysis systemizes the findings in the literature for the one-compartment TMDD model and provides some new insights about the control of the process. These findings are very important in order to i) propose improvements in the design of new TMDD models and ii) find ways to control the evolution of the process on existing TMDD models by identifying the correct parameters that must be more accurately specified [3].
References:
[1] Lam, S. H., & Goussis, D. A. The CSP method for simplifying kinetics. International Journal of Chemical Kinetics, 26(4), 461-486, 1994.
[2] Goussis, D. A. Quasi steady state and partial equilibrium approximations: their relation and their validity. Combustion Theory and Modelling, 16(5), 869-926, 2012.
[3] Patsatzis, Dimitris G., Dimitris T. Maris, and Dimitris A. Goussis. "Asymptotic analysis of a target-mediated drug disposition model: algorithmic and traditional approaches." Bulletin of mathematical biology 78.6 (2016): 1121-1161.
