Theoretical Considerations of Target-Mediated Drug Disposition Models: Simplifications and Approximations

ABSTRACT

Purpose

To clarify relationships among various types of target-mediated disposition (TMD) models including the Michaelis-Menten, quasi-steady-state (Qss), and rapid binding models and propose measures for the closeness of some models as approximations to the general TMD model (Mager and Jusko, J Pharmacokinet Pharmacodyn 28(6):507–532, 2001).

Methods

Based on the classic singular perturbation theory by selecting appropriate scales of time, we derive requirements with which the Michaelis-Menten and Qss models are suitable approximations. Under the Qss assumption we show that other simplifications of the general TMD model can be similarly obtained as the Michaelis-Menten and Qss models. We compare these models by simulations using known application examples.

Results

The Michaelis-Menten and Qss models are direct simplifications of the general TMD model and, moreover, suitable approximations if certain specific requirements on the parameters are met.

Conclusions

As a first attempt to quantify the closeness of some simplifications to the general TMD model, our work should provide a more rigorous basis for the theoretical and practical research of TMD models, which are important for investigating the pharmacokinetic-pharmacodynamic relationships of many biological compounds.

APPENDIX

The one-compartment Michaelis-Menten model with a bolus injection and constant total target is a particularly illustrative case for a simplification of the general TMD model under the Qss assumption. We provide here more details of the free drug C that can be expressed using a special mathematical function and the complex M that can be improved over the usual choice. We recommend the use of the improved expression of M (Eq. 46) in modeling.

Inner and Outer Layers

From the terminology of singular perturbation theory, the free drug C (≈ C ₀) and complex M given in (29) as a solution to (28) are valid at the initial fast binding time and called the inner layer, but the solution of C and M to (30) and (31) are valid at the later quasi-steady state and called the outer layer. If k _e > 0, Eq. 30 for the outer layer can be transformed to a (transcendental) equation after integration

$$ \frac{C}{{{K_{\text{m}}}}}{\left( {1 + \frac{{{k_{\text{e}}}}}{{{k_{\text{e}}} + {k_{\text{n}}}}}\frac{C}{{{K_{\text{m}}}}}} \right)^{{{k_{\text{n}}}/{k_{\text{e}}}}}} = {G_0}{e^{{ - ({k_{\text{e}}} + {k_{\text{n}}})t}}}, $$

(45)

where ${k_{\text{n}}} = {V_{{\max }}}/{K_{\text{m}}}$ and ${V_{{\max }}} = {k_{\text{met}}}{R_{\text{tot}}}$. The parameter k _n is the maximum nonlinear elimination rate constant for the free drug in the Michaelis-Menten model. The constant ${G_0} = {G_0}\left( {{C_0}/{K_{\text{m}}},\;{k_{\text{n}}}/{k_{\text{e}}}} \right)$ is equal to the left-hand side of the above equation at t = 0 and thus dependent on C ₀/K _m and k _n/k _e. If k _e = 0, then V _max > 0 (see the assumption of nontrivial TMD models in “Methods”), and the integration of (30) results in the limit of the above equation as k _e → 0:

$$ \left( {C/{K_{\text{m}}}} \right){e^{{C/{K_{\text{m}}}}}} = {G_0}{e^{{ - {k_{\text{n}}}t}}}. $$

Thus, C is equal to ${K_{\text{m}}}W\left( {{G_0}{e^{{ - {k_{\text{n}}}t}}}} \right)$ at k _e = 0, where W is the Lambert W-function (or omega function), which is the inverse of f (x) = xe^x. As a generalization of W, if we denote W(x; α) as the inverse of $f\left( {x;\alpha } \right) = x{\left( {1 + \alpha x} \right)^{{1/\alpha }}}$ then the free drug concentration C in (45) is approximately ${K_{\text{m}}}W\left( {{G_0}{e^{{\left( { - {k_{\text{e}}} + {k_{\text{n}}}} \right)t}}};\;{k_{\text{e}}}/\left( {{k_{\text{e}}} + {k_{\text{n}}}} \right)} \right)$ for small k _e.

Note that a consequence of making the Qss assumption is M (0) = M ₀ ≠ 0 if M is the outer layer and given by the first formula in (13), but the limit of the outer layer as t → 0 is the same as the inner layer in (29) as t → ∞. Thus, an improvement to both inner and outer layers as approximations of the complex can be made. With a standard matching technique, a total solution for M can be proposed as the sum of the inner and outer layers minus the overlapping term, which gives

$$ M = {R_{\text{tot}}}\left[ {\frac{C}{{{K_{\text{m}}} + C}} - \frac{{{C_0}}}{{{K_{\text{m}}} + {C_0}}}\exp ( - {k_{\text{on}}}({K_{\text{m}}} + {C_0})t)} \right], $$

(46)

where C is given in (45). Simulation examples in Figs. 5 and 6 show that the improvement corrects the over-estimation of the complex at initial time. In all cases, for small t the improved complex M overlaps completely with that of the general TMD model (plots not shown).