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Numerical validation and properties of a rapid binding approximation of a target-mediated drug disposition pharmacokinetic model

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Abstract

Target mediated drug disposition (TMDD) describes the phenomenon where high affinity binding of a drug to its pharmacological target (enzymes or receptors) significantly alters the pharmacokinetic profile of the drug. A rapid binding model replaces the often inestimable binding micro-constants (k on and k off) of TMDD models with the equilibrium dissociation constant (K D) by assuming rapid binding of the drug to its target. The purpose of this study is to examine the validity of the rapid binding assumption and the pharmacokinetic properties of this model. Temporal profiles of free drug in plasma and a non-specific distribution site, free receptor, and the pharmacodynamically relevant, drug–receptor complex obtained from the rapid binding model compared favorably with the full TMDD model for small values of the parameter ɛ, which represents the ratio of the time required for drug–receptor binding relative to the time required for drug to be cleared from the system. The effect of escalating drug doses on the temporal characteristics and the comparison between the two models has been numerically investigated. A closer match between the full and rapid binding models is observed for high doses. Analysis for very large doses (Dose/V c) relative to endogenous steady-state receptor concentration (R ss), reveals that the rapid binding model reduces to a standard two compartmental model with a plasma compartment with linear drug elimination and a peripheral compartment. Decreasing clearance with increasing dose and decreasing R ss indicates that for drugs exhibiting TMDD, the relative ratio of R ss and dose is an important determinant of the pharmacokinetic properties rather than the individual parameters alone. An analytical solution derived for clearance shows that the primary elements of the apparent clearance of the drug are the linear clearance given by k el V c, the non-linear clearance due to drug–receptor complex internalization (k int), and the ratio of AUC values of the receptor complex to that of free drug. Overall, simulations and analytical techniques applied here provide a better understanding of the validity of the rapid binding model and provide guidelines for its application.

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Acknowledgements

This study was supported by Grant GM57980 (for D.E.M. and W.K.) and funds for a postdoctoral fellowship from Amgen, Inc. (to A.M.).

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Correspondence to Donald E. Mager.

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Appendices

Appendix 1

Derivation of apparent clearance for the rapid binding model

In order to obtain an analytical solution for clearance of the rapid binding model, Eqs. 9 and 10 are integrated:

$$ \int\limits_{0}^{\infty } {\frac{{{\text{d}}C_{\text{tot,rb}} }}{{{\text{d}}t}}{\text{d}}t} = \int\limits_{0}^{\infty } {\left( { - k_{\text{int}} C_{\text{tot,rb}} - \left( {k_{\text{el}} + k_{\text{pt}} - k_{\text{int}} } \right)C_{\text{rb}} + k_{\text{tp}} \frac{{A_{\text{T,rb}} }}{{V_{\text{c}} }}} \right)} {\text{d}}t $$
(33)
$$ \int\limits_{0}^{\infty } {\frac{{{\text{d}}A_{\text{T, rb}} }}{{{\text{d}}t}}} {\text{ d}}t = \int\limits_{0}^{\infty } {\left( {k_{\text{pt}} C_{\text{rb}} V_{\text{c}} - k_{\text{tp}} A_{\text{T,rb}} } \right)} {\text{d}}t $$
(34)

Multiplying Eq. 33 by V c and adding to Eq. 34 yields the following relation:

$$ {\text{Dose}} = k_{\text{el}} V_{\text{c}} \int\limits_{0}^{\infty } {C_{\text{rb}} {\text{d}}t} + k_{\text{int}} V_{\text{c}} \int\limits_{0}^{\infty } {\left( {C_{\text{tot,rb}} - C_{\text{rb}} } \right)} {\text{d}}t $$
(35)

Here, \( \int\nolimits_{0}^{\infty } {C_{\text{rb}} {\text{d}}t} \) represents the area under the plasma concentration-time curve of the free drug \( \left( \left({AUC} \right)_{\text{Crb}}\right) \). Substituting Eq. 7, in \( \int\nolimits_{0}^{\infty } {\left( {C_{\text{tot,rb}} - C_{\text{rb}} } \right)} {\text{d}}t \) yields the term,\( \int\nolimits_{0}^{\infty } {\left( {RC_{\text{rb}} } \right)} {\text{d}}t \), which represents the area under the concentration-time curve of the drug–receptor complex \( \left( \left({AUC} \right)_{\text{RCrb}}\right) \). Dividing Eq. 35 by the \( \left( {AUC} \right)_{\text{Crb}} \) gives the apparent clearance of the target mediated drug as:

$$ CL = k_{\text{el}} V_{\text{c}} + k_{\text{int}} V_{\text{c}} \frac{{\left( {AUC} \right)_{\text{RCrb}} }}{{\left( {AUC} \right)_{\text{Crb}} }} $$
(36)

Appendix 2

Large dose approximation of the rapid binding model

It is important to understand the limiting behavior of the rapid binding model for very large dose level compared to the endogenous receptor concentration. For the limiting case of very large dose relative to R ss (= k syn/k deg), the dimensionless parameter, δ given by Eq. 21 is a small parameter. Mathematically, this is represented as:

$$ \delta = \frac{{k_{\text{syn}} }}{{k_{ \rm deg } \left( {{{{\text{Dose}}} \mathord{\left/ {\vphantom {{{\text{Dose}}} {V_{\text{c}} }}} \right. \kern-\nulldelimiterspace} {V_{\text{c}} }}} \right)}} \ll 1 $$
(37)

Eqs. 23 and 29 imply that

$$ rc_{\rm rb} = \frac{{r_{\rm {tot,rb}} c_{\rm rb} }}{{\lambda + c_{\rm rb} }} $$
(38)

Consequently, Eq. 26 can be transformed to a δ free form

$$ \frac{{{\text{d}}r_{\text{tot, rb}} }}{{{\text{d}}\tau }} = \kappa - \frac{{\left( {\mu - \kappa } \right)r_{\rm {tot,rb}} c_{rb} }}{{\lambda + c_{\rm rb} }} - \kappa r_{\text{tot, rb}} $$
(39)

Since the solution of the system of differential Eqs. 2326 continuously depend on δ, one can obtain the large dose approximation by letting δ → 0. Thus, Eqs. 27 and 28 show that the free drug concentration is equal to the total drug concentration

$$ c_{\text{tot, rb}} = c_{\text{rb}} $$
(40)

Substituting Eq. 40 in Eqs. 24 and 25 results in the following set of differential equations that represent the governing equations of a two compartmental linear model with linear elimination only from the central compartment:

$$ \frac{{{\text{d}}c_{\text{rb}} }}{{{\text{d}}\tau }} = - \left( {1 + \beta } \right)c_{\text{rb}} + \gamma a_{\text{T, rb}} $$
(41)
$$ \frac{{{\text{d}}a_{\text{T, rb}} }}{{{\text{d}}\tau }} = \beta c_{\text{rb}} - \gamma a_{\text{T, rb}} $$
(42)

The clearance (CL), and steady-state volume of distribution (V ss) of a two compartmental linear system has been well established [19] and in dimensional form, they are as follows:

$$ CL = V_{\text{c}} k_{\text{el}} $$
(43)
$$ V_{\text{ss}} = V_{\text{c}} \left( {1 + \frac{{k_{\text{pt}} }}{{k_{\text{tp}} }}} \right) $$
(44)

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Marathe, A., Krzyzanski, W. & Mager, D.E. Numerical validation and properties of a rapid binding approximation of a target-mediated drug disposition pharmacokinetic model. J Pharmacokinet Pharmacodyn 36, 199–219 (2009). https://doi.org/10.1007/s10928-009-9118-8

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