Asymptotic analysis of a TMDD model: when a reaction contributes to the destruction of its product
Target-mediated drug disposition (TMDD) refers to the interaction, \textit{in vivo}, between a drug and a biological receptor which subsequently form a drug-receptor complex. That complex represents the active configuration of the drug and is crucial to both the pharmacodynamics and the pharmacokinetics thereof. A modelling framework for TMDD was established in [\textit{D. E. Mager} and \textit{W. J. Jusko}, ``General pharmacokinetic model for drugs exhibiting target-mediated drug disposition'', J. Pharmacokinet. Pharmacodyn. 28, No. 6, 507--532 (2001; \url{doi:10.1023/a:1014414520282})]; their differential equation model consists of a central compartment (blood plasma) and a peripheral compartment, the tissue, and assumes first-order absorption from a depot compartment. Binding between the drug, or ligand, and the receptor (target) occurs in the central compartment; ligand is allowed to move between compartments. Subsequent modifications of that standard model included varying numbers of compartments, the absence of absorption, the presence of multiple drugs or receptors, and autoregulation; a review can be found in [\textit{P. Dua}, \textit{E. Hawkins} and \textit{P. H. van der Graaf}, ``A tutorial on target-mediated drug disposition (TMDD) models'', Pharmacomet. Syst. Pharmacol. 4, No. 6, 324--337 (2015; \url{doi:10.1002/psp4.41})]. Common to these models is a time scale separation which is due to ligand-receptor binding being fast, while the elimination of ligand is slow [\textit{L. Gibiansky} et al., ``Approximations of the target-mediated drug disposition model and identifiability of model parameters'', J. Pharmacokin. Pharmacodyn. 35, No. 5, 573--591 (2008; \url{doi:10.1007/s10928-008-9102-8})]; that separation prompted the construction of reduced models on the basis of a quasi-steady-state approximation (QSSA) or a partial equilibrium approximation (PEA) [Dua, Hawkins and van der Graaf, loc. cit.]. \par Recently, \textit{L. A. Peletier} and \textit{J. Gabrielsson} [``Dynamics of target-mediated drug disposition: characteristic profiles and parameter identification'', J. Pharmacokinet. Pharmacodyn. 39, No. 5, 429--451 (2012; \url{doi:10.1007/s10928-012-9260-6})] identified four distinct phases, which characterise different states of drug disposition, in a one-compartment TMDD model. Here, the authors investigate whether the same phases are observed in a two-compartment model with first-order absorption, in generalisation of prior work by \textit{D. G. Patsatzis} et al. [Bull. Math. Biol. 78, No. 6, 1121--1161 (2016; Zbl 1348.92075)]. \par The authors' analysis is based on the algorithmic methodology of Computational Singular Perturbation (CSP) [\textit{S. H. Lam} and the second author, ``The CSP method for simplifying kinetics'', Int. J. Chem. Kinet. 26, No. 4, 461--486 (1994; \url{doi:10.1002/kin.550260408})]. In the process, they show that the dynamics of their two-compartment TMDD model evolves mainly on two low-dimensional manifolds in phase space, interspersed with rapid transient phases, in analogy to the corresponding one-compartment model. Moreover, they assess in detail the dependence of that dynamics on various rate constants; in particular, they clarify the counterintuitive observation that an increase in the rate of transfer of ligand from tissue to the central compartment increases the depletion rate of ligand. Finally, and interestingly, they argue that a traditional QSSA or PEA may not always lead to a valid reduction in the model considered here.
computational singular perturbation (CSP)
scientific article; zbMATH DE number 7031407
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