Development of a Complex Parent-Metabolite Joint Population Pharmacokinetic Model

Abstract

This study aimed to develop a joint population pharmacokinetic model for an antipsychotic agent in development (S33138) and its active metabolite (S35424) produced by reversible metabolism. Because such a model leads to identifiability problems and numerical difficulties, the model building was performed using the FOCE-I and the Stochastic Approximation Expectation Maximization (SAEM) estimation algorithms in NONMEM and MONOLIX, respectively. Four different structural models were compared based on Bayesian information criteria. Models were first written as ordinary differential equations systems and then in closed form (CF) to facilitate further analyses. The impact of polymorphisms on genes coding for the CYP2C19 and CYP2D6 enzymes, respectively involved in the parent drug and the metabolite elimination were investigated using permutation Wald test. The parent drug and metabolite plasma concentrations of 101 patients were analyzed on two occasions after 4 and 8 weeks of treatment at 1, 3, 6, and 24 h following daily oral administration. All configurations led to a two compartment model with back-transformation of the metabolite into the parent drug and a first-pass effect. The elimination clearance of the metabolite through other processes than back-transformation was decreased by 35% [9–53%] in CYP2D6 poor metabolizer. Permutation tests were performed to ensure the robustness of the analysis, using SAEM and CF. In conclusion, we developed a complex joint pharmacokinetic model adequately predicting the impact of CYP2D6 polymorphisms on the parent drug and its metabolite concentrations through the back-transformation mechanism.

APPENDIX

In all the equations below, C _p is the parent drug concentration in plasma and C _m is the active metabolite concentration in plasma following a single oral administration of a dose D of the parent drug.

The ordinary differential equation system and the corresponding closed-form solution correspond to the last model on the right-hand side of Fig. 2, with similar absorption rates K _a = K _ap = K _am.

Ordinary Differential Equation System

$$ \begin{array}{*{20}{c}} {\frac{{d{C_p}}}{{dt}} = \frac{{{K_a}fD\,{F_p}\,{e^{{ - {K_a}t}}}}}{{{V_p}}} - \left( {{K_{{po}}} + {K_{{pm}}}} \right){C_p} + {k_{{mp}}}\frac{{{V_m}}}{{{V_p}}}{C_m}} \\ {\frac{{d{C_m}}}{{dt}} = \frac{{{K_a}fD\,\,\left( {1 - {F_p}} \right)\,{e^{{ - {K_a}t}}}}}{{{V_m}}} - \left( {{k_{{mo}}} + {k_{{mp}}}} \right){C_m} + {k_{{pm}}}\frac{{{V_p}}}{{{V_m}}}{C_p}} \\ \end{array} $$

(4)

In the ordinary differential equation system (4), f is the fraction of dose after absorption, D is the dose, F _p is the fraction of parent reaching systemic circulation after absorption, K _a is the absorption constant for the parent and the metabolite, V _p is the volume of distribution for the parent, V _m is the volume of distribution for the metabolite, k _po is the parent rate constant of elimination by other pathways (=CL _po/V _p), k _pm is the parent rate constant of transformation into the metabolite (=CL _pm/V _p), k _mo is the metabolite rate constant of elimination by other pathways (=CL _mo/V _m), and k _mp is the metabolite rate constant of back-transformation into the parent (=CL _mp/V _m).

Closed-Form Solutions

$$ \begin{array}{*{20}{c}} {{C_p} = \frac{{fD{K_a}\left( {\left( {{E_2} - {K_a}} \right) + {k_{{mp}}}\left( {1 - {F_p}} \right)/{F_p}} \right)}}{{\left( {{V_p}/{F_p}} \right)\left( {{\lambda_1} - {K_a}} \right)\left( {{\lambda_2} - {K_a}} \right)}}{e^{{ - {K_a}t}}} + \frac{{fD{K_a}\left( {\left( {{E_2} - {\lambda_1}} \right) + {k_{{mp}}}\left( {1 - {F_p}} \right)/{F_p}} \right)}}{{\left( {{V_p}/{F_p}} \right)\left( {{K_a} - {\lambda_1}} \right)\left( {{\lambda_2} - {\lambda_1}} \right)}}{e^{{ - {\lambda_1}t}}}} \hfill \\ { + \frac{{fD{K_a}\left( {\left( {{E_2} - {\lambda_2}} \right) + {k_{{mp}}}\left( {1 - {F_p}} \right)/{F_p}} \right)}}{{\left( {{V_p}/{F_p}} \right)\left( {{K_a} - {\lambda_2}} \right)\left( {{\lambda_1} - {\lambda_2}} \right)}}{e^{{ - {\lambda_2}t}}}} \hfill \\ \end{array} $$

(5)

$$ \begin{array}{*{20}{c}} {{C_m} = \frac{{fD{K_a}\left( {\left( {{E_1} - {K_a}} \right) + {k_{{pm}}}{F_p}/\left( {1 - {F_p}} \right)} \right)}}{{\left( {{V_m}/\left( {1 - {F_p}} \right)} \right)\left( {{\lambda_1} - {K_a}} \right)\left( {{\lambda_2} - {K_a}} \right)}}{e^{{ - {k_a}t}}} + \frac{{fD{K_a}\left( {\left( {{E_1} - {\lambda_1}} \right) + {k_{{pm}}}{F_p}/\left( {1 - {F_p}} \right)} \right)}}{{\left( {{V_m}/\left( {1 - {F_p}} \right)} \right)\left( {{K_a} - {\lambda_1}} \right)\left( {{\lambda_2} - {\lambda_1}} \right)}}{e^{{ - {\lambda_1}t}}}} \hfill \\ { + \frac{{fD{K_a}\left( {\left( {{E_1} - {\lambda_2}} \right) + {k_{{pm}}}{F_p}/\left( {1 - {F_p}} \right)} \right)}}{{\left( {{V_m}/\left( {1 - {F_p}} \right)} \right)\left( {{K_a} - {\lambda_2}} \right)\left( {{\lambda_1} - {\lambda_2}} \right)}}{e^{{ - {\lambda_2}t}}}} \hfill \\ \end{array} $$

(6)

In both Eqs. (5) and (6), E ₁ is the parent drug total rate constant of elimination (=k _po + k _pm), E ₂ is the metabolite total rate constant of elimination (=k _mo + k _mp), and λ ₁ and λ ₂ are the initial and terminal slopes of elimination, respectively defined in Eqs. (7) and (8).

$$ {\lambda_1} = \frac{{\left( {{E_1} + {E_2}} \right) + \sqrt {{{{\left( {{E_1} + {E_2}} \right)}^2} - 4\left( {{E_1}{E_2} - {k_{{mp}}}{k_{{pm}}}} \right)}} }}{2} $$

(7)

$$ {\lambda_2} = \frac{{\left( {{E_1} + {E_2}} \right) + \sqrt {{{{\left( {{E_1} + {E_2}} \right)}^2} - 4\left( {{E_1}{E_2} - {k_{{mp}}}{k_{{pm}}}} \right)}} }}{2} $$

(8)

Parameter Identifiability

From the model slopes, the following parameters may be identified: K _a, λ ₁, and λ ₂. From the model intercepts, we can identify the following equations:

$$ V = \frac{{\left( {{E_2} - {\lambda_1}} \right) + {k_{{mp}}}\left( {1 - {F_p}} \right)/{F_p}}}{{{V_p}/f{F_p}}} $$

(9)

$$ W = \frac{{\left( {{E_2} - {\lambda_2}} \right) + {k_{{mp}}}\left( {1 - {F_p}} \right)/{F_p}}}{{{V_p}/f{F_p}}} $$

(10)

$$ Y = \frac{{\left( {{E_1} - {\lambda_1}} \right) + {k_{{mp}}}{F_p}/\left( {1 - {F_p}} \right)}}{{{V_m}/f\left( {1 - {F_p}} \right)}} $$

(11)

$$ Z = \frac{{\left( {{E_1} - {\lambda_2}} \right) + {k_{{mp}}}{F_p}\left( {1 - {F_p}} \right)}}{{{V_m}/f\left( {1 - {F_p}} \right)}} $$

(12)

From (7) and (8) we can write:

$$ {\lambda_1} + {\lambda_2} = {E_1} + {E_2} $$

(13)

$$ {\lambda_1}{\lambda_2} = {E_1}{E_2} - {k_{{pm}}}\left( {{F_p}/\left( {1 - {F_p}} \right)} \right){k_{{mp}}}\left( {\left( {1 - {F_p}} \right)/{F_p}} \right) $$

(14)

This yields a system of six equations, where V, W, Y, and Z are four reals like λ ₁ and λ ₂. We can estimate the following parameters: V _p/f F _p, V _m /f (1 − F _p), E ₁ = k _po + k _pm, E ₂ = k _mo + k _mp, k _pm F _p/(1 − F _p) and k _mp(1 − F _p)/F _p, with the following solutions:

$$ \begin{array}{*{20}{c}} {{E_1} = \left( {{\lambda_1} + {\lambda_2} - {\lambda_1}{\lambda_2}/{T_2} - {T_1}} \right)/\left( {1 - {T_1}/{T_2}} \right)} \hfill \\ {{E_2} = {\lambda_1} + {\lambda_2} - {E_1}} \hfill \\ {{k_{{mp}}}\left( {1 - {F_p}} \right)/{F_p} = \left( {{\lambda_2} - {\lambda_1}\left( {W/V} \right)} \right)/\left( {1 - W/V} \right) - {E_2}} \hfill \\ {{k_{{mp}}}{F_p}/\left( {1 - {F_p}} \right) = \left( {{\lambda_2} - {\lambda_1}\left( {Z/Y} \right)} \right)/\left( {1 - Z/Y} \right) - {E_1}} \hfill \\ {{V_p}/f{F_p} = \frac{{\left( {{E_2} - {\lambda_1}} \right) + {k_{{mp}}}\left( {1 - {F_p}} \right)/{F_p}}}{V}} \hfill \\ {{V_m}/f\left( {1 - {F_p}} \right) = \frac{{\left( {{E_1} - {\lambda_1}} \right) + {k_{{pm}}}{F_p}/\left( {1 - {F_p}} \right)}}{Y}} \hfill \\ \end{array} $$

(15)

where ${T_1} = \left( {\frac{{{\lambda_2}}}{W} - \frac{{{\lambda_1}}}{V}} \right)/\left( {\frac{1}{W} - \frac{1}{V}} \right)$ and ${T_2} = \left( {\frac{{{\lambda_2}}}{Z} - \frac{{{\lambda_1}}}{Y}} \right)/\left( {\frac{1}{Z} - \frac{1}{Y}} \right)$. Alternatively, the following parameterization may be used instead:

$$ \begin{array}{*{20}{c}} {{V_p}/f{F_p}} \hfill \\ {{V_m}/f\left( {1 - {F_p}} \right)} \hfill \\ {C{L_{{ptot}}}/f{F_p} = {E_1} \times {V_p}/f{F_p}} \hfill \\ {C{L_{{mtot}}}/f\left( {1 - {F_p}} \right) = {E_2} \times {V_m}/f\left( {1 - {F_p}} \right)} \hfill \\ {C{L_{{pm}}}/{V_m} = {k_{{pm}}}{V_p} = {k_{{pm}}}{F_p}/\left( {1 - {F_p}} \right) \times {V_p}/f{F_p} \times f\left( {1 - {F_p}} \right)/{V_m}} \hfill \\ {C{L_{{mp}}}/{V_p} = {k_{{mp}}}{V_m}/{V_p} = {k_{{mp}}}\left( {1 - {F_p}} \right)/{F_p} \times {V_m}/f\left( {1 - {F_p}} \right) \times f{F_p}/{V_p}} \hfill \\ \end{array} $$

(16)