Impact of saturable distribution in compartmental PK models: dynamics and practical use

Abstract

We explore the impact of saturable distribution over the central and the peripheral compartment in pharmacokinetic models, whilst assuming that back flow into the central compartiment is linear. Using simulations and analytical methods we demonstrate characteristic tell-tale differences in plasma concentration profiles of saturable versus linear distribution models, which can serve as a guide to their practical applicability. For two extreme cases, relating to (i) the size of the peripheral compartment with respect to the central compartment and (ii) the magnitude of the back flow as related to direct elimination from the central compartment, we derive explicit approximations which make it possible to give quantitative estimates of parameters. In three appendices we give detailed explanations of how these estimates are derived. They demonstrate how singular perturbation methods can be successfully employed to gain insight in the dynamics of multi-compartment pharmacokinetic models. These appendices are also intended to serve as an introductory tutorial to these ideas.

Keywords: Distribution; Pharmacokinetics; Saturation.

Fig. 1

Individual plasma concentration versus time profiles for six subjects receiving a once-daily oral 1500 mg over a period of 3 weeks. The cyan dots show the observed plasma concentrations, the black curve shows the individual fit and the grey curve the population fit of the 2-receptor model, while the magenta curves show the individual fits of the 1-receptor model

Fig. 2

Linear model (2) graphs of $A_2(t)$ for increasing infusion rates q $=$ 1, 2, 3, 4, 5 mg h${}^{-1}$ when $k_a=10$, $k_{20}={10}^{-2}$, $k_p={10}^{-4}$ h${}^{-1}$ and $H=100$

Fig. 3

Nonlinear model (3) graphs of $A_2$ versus time for the parameter values $k_a=10$, $k_{20}={10}^{-2}$, $k_p={10}^{-4}$ h${}^{-1}$, $B_{\max }=3\times {10}^4$ mg, $K_M={10}^2$ mg

Fig. 4

Nonlinear model (3) graphs of $A_2$ versus time for $D_0=10,20,\dots ,70$ for the parameter values $k_a=5$, $k_{20}=0.2$ h${}^{-1}$, $k_p=1$ h${}^{-1}$, $B_{\max }=100$ mg, $K_M=10$ mg

Fig. 5

Variation of the plateau value ${\overline{A}}_2(q)$ (left) and the normalised plateau value ${\overline{A}}_2(q)/q$ (right) for the nonlinear model as they vary with q, when the data are $k_p={10}^{-4}$ h${}^{-1}$, $k_{20}={10}^{-2}$ h${}^{-1}$, the capacity takes the values: $B_{\max }={10}^4$ (blue) $3\times {10}^4$ (red) and $6\times {10}^4$ (green) mg, and $K_M=100$ mg

Fig. 6

Terminal slopes: ${\mathit{\lambda }}_z^{(1)}(q,B_{\max })$ (left) in the first phase and ${\mathit{\lambda }}_z^{(2)}(q,B_{\max })$ (right) in the second phase versus the infusion rate q for the nonlinear model for two values of the capacity: $B_{\max }={10}^4$ (red) and $B_{\max }=3\times {10}^4$ mg (blue) and the rate constants $k_a=10$, $k_{20}={10}^{-2}$, $k_p={10}^{-4}$ h${}^{-1}$, and $K_M={10}^2$ mg

Fig. 7

Nonlinear model with leakage from the peripheral compartment (3) & (30). Graphs of $A_2$ versus time for $q=5$ and $\mathit{\alpha }=0,0.5,1,2,4$ for the parameter values $k_a=10$ h${}^{-1}$, $k_{20}=0.01$ h${}^{-1}$, $k_p={10}^{-4}$ h${}^{-1}$, $B_{\max }=3\times {10}^4$ mg, $K_M={10}^2$ mg