Pharmacokinetics and Modeling of Quercetin and Metabolites

Purpose

To determine the pharmacokinetics of quercetin and its glucuronide/sulfate conjugates and to develop a pharmacokinetic model to simultaneously describe their disposition after intravenous and oral administration in rats.

Methods

After oral, intraportal, and intravenous administration of quercetin, serial plasma, urine, and fecal concentrations of quercetin and its conjugates were determined by an HPLC method. Enterohepatic recirculation was evaluated in a linked-rat model as well as after oral administration of bile containing quercetin and its metabolites. Based on the experimental data, a specific compartmental model was developed and validated to describe and predict the plasma concentration-time profiles of quercetin and its conjugates after oral and intravenous administration.

Results

Only 5.3% of unchanged quercetin was bioavailable, although the total quercetin absorbed was as high as 59.1%. After oral administration, about 93.3% of quercetin was metabolized in the gut, with only 3.1% metabolized in the liver. No significant enterohepatic recirculation was observed for both quercetin and its conjugated metabolites. The pharmacokinetic model fitted well the observed data of quercetin and its conjugates.

Conclusions

Our study clarifies the relative importance of the gut, liver, and bile in the metabolism and excretion of quercetin and its conjugates. The pharmacokinetic model appears to be suitable for describing the absorption and disposition of the quercetin and its conjugates and may be applicable to other flavonoids that undergo similar pharmacokinetic pathways.

Mass Balance of Each Compartment

The differential equations for the change of amount ${\left( {\frac{{dX}} {{dt}}} \right)}$ of substance with time in each compartment following an iv dose are shown as follows:

$$ \frac{{dX_{1} }} {{dt}} = - {\left( {K_{{12}} + K_{f} + K_{{10}} } \right)}X_{1} + K_{{21}} X_{2} $$

(A1)

$$ \frac{{dX_{2} }} {{dt}} = K_{{12}} X_{1} - K_{{21}} X_{2} $$

(A2)

$$ \frac{{dX_{3} }} {{dt}} = K_{f} X_{1} - {\left( {K_{{34}} + K_{{30}} } \right)}X_{3} + K_{{43}} X_{4} $$

(A3)

$$ \frac{{dX_{4} }} {{dt}} = K_{{34}} X_{3} - K_{{43}} X_{4} $$

(A4)

where $\frac{{dX_{1} }} {{dt}}$, $\frac{{dX_{2} }} {{dt}}$, $\frac{{dX_{3} }} {{dt}}$, and $\frac{{dX_{4} }} {{dt}}$ are the change of amount of substance with time in compartments 1, 2, 3, and 4 respectively, and other terms have been described previously.

The differential equations for this model with an oral dose are shown as follows:

$$ \frac{{dX_{{g1}} + dX_{{g2}} }} {{dt}} = - {\left( {K_{{a1}} X_{{g1}} + K_{{a2}} X_{{g2}} } \right)} $$

(A5)

$$ \frac{{dX_{1} }} {{dt}} = K_{{a1}} X_{{g1}} - {\left( {K_{{12}} + K_{f} + K_{{10}} } \right)}X_{1} + K_{{21}} X_{2} $$

(A6)

$$ \frac{{dX_{2} }} {{dt}} = K_{{12}} X_{1} - K_{{21}} X_{2} $$

(A7)

$$ \frac{{dX_{3} }} {{dt}} = K_{{a2}} X_{{g2}} + K_{f} X_{1} - {\left( {K_{{34}} + K_{{30}} } \right)}X_{3} + K_{{43}} X_{4} $$

(A8)

$$ \frac{{dX_{4} }} {{dt}} = K_{{34}} X_{3} - K_{{43}} X_{4} $$

(A9)

where $\frac{{dX_{{g1}} }} {{dt}}$ and $\frac{{dX_{{g2}} }} {{dt}}$ are the change of amount of substance with time in the gut for the parent drug and metabolites to be absorbed, so that at time 0, X_g1 and X_g2 equal to F₁D₀ and F₂D₀ respectively (see Fig. 1); all other terms have been defined previously.

Model Identifiability Analysis

To test the validity of this model (i.e. to test whether the model parameters can be practically identified), structural identifiability analysis was performed using the similarity transformation approach (32).

Generally, a linear system can be described as:

$$ \frac{{dx}} {{dt}} = A{\left( p \right)}x + B{\left( p \right)}u,x{\left( {t_{0} } \right)} = x_{0} {\left( p \right)} $$

(A10)

$$ y = C{\left( p \right)}x $$

(A11)

where A is the (n × n) system matrix, B is the input matrix, and C is the output matrix, x₀ represents the initial condition and p represents the unknown parameter in these matrices.

The similarity transformation approach utilizes the fact that for another model characterized as ${\left( {\overline{{A,}} \overline{B} ,\overline{C} } \right)}$, it is necessary and sufficient that there exists a nonsingular matrix T such that:

$$ T\overline{B} = B $$

(A12)

$$ \overline{C} = CT $$

(A13)

$$ T\overline{A} = AT $$

(A14)

That is, if solving Eqs. (A12)–(A14) yields the solution that T is the n × n identity matrix, the model is then globally identifiable.

For the model described in Fig. 1, structural identifiability analysis was performed under the condition that both iv and oral doses of quercetin were administered, and the plasma concentrations of unchanged quercetin and its combined glucuronide/sulfate conjugates were measured.

The matrixes of the model are as follows:

$$ A = {\left( {\begin{array}{*{20}c} {{ - {\left( {K_{{a1}} + K_{{a2}} } \right)}}} & {0} & {0} & {0} & {0} \\ {{K_{{a1}} }} & {{ - {\left( {K_{{12}} + K_{f} + K_{{10}} } \right)}}} & {{K_{{21}} }} & {0} & {0} \\ {0} & {{K_{{12}} }} & {{ - K_{{21}} }} & {0} & {0} \\ {{K_{{a2}} }} & {{K_{f} }} & {0} & {{ - {\left( {K_{{34}} + K{}_{{30}}} \right)}}} & {{K_{{43}} }} \\ {0} & {0} & {0} & {{K_{{34}} }} & {{ - K_{{43}} }} \\ \end{array} } \right)} $$

(A15)

$$ B = {\left( {\begin{array}{*{20}c} {1} & {0} \\ {0} & {1} \\ {0} & {0} \\ {0} & {0} \\ {0} & {0} \\ \end{array} } \right)} $$

(A16)

$$ C = {\left( {\begin{array}{*{20}c} {0} & {{\frac{1} {{V_{1} }}}} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} & {{\frac{1} {{V_{3} }}}} \\ \end{array} } \right)} $$

(A17)

where V₁ and V₃ are the volumes of compartment 1 and 3, respectively.

Let P be the vector of the unknown parameters, and Tbe a 5 × 5 matrix $P \in {\left[ {K_{{a1}} ,K_{{a2}} ,V_{1} ,V_{3} ,K_{{12}} ,K_{{21}} ,K_{{10}} ,K_{f} ,K_{{34}} ,K_{{43}} ,K_{{30}} } \right]}$

$$ T = {\left( {\begin{array}{*{20}c} {{t_{{11}} }} & {{t_{{12}} }} & {{t_{{13}} }} & {{t_{{14}} }} & {{t_{{15}} }} \\ {{t_{{21}} }} & {{t_{{22}} }} & {{t_{{23}} }} & {{t_{{24}} }} & {{t_{{25}} }} \\ {{t_{{31}} }} & {{t_{{32}} }} & {{t_{{33}} }} & {{t_{{34}} }} & {{t_{{35}} }} \\ {{t_{{41}} }} & {{t_{{42}} }} & {{t_{{43}} }} & {{t_{{44}} }} & {{t_{{45}} }} \\ {{t_{{51}} }} & {{t_{{52}} }} & {{t_{{53}} }} & {{t_{{54}} }} & {{t_{{55}} }} \\ \end{array} } \right)} $$

(A18)

By substituting A, B, C, and T into Eqs. (A12)–(A14) gives the following results: 1) V₁, K_a1, K₁₂, K₂₁, K₃₄, K₄₃, and K₃₀ are identifiable, whereas K_a2, K _f , K_1e and V₃ are not identifiable; 2) if V₃ is known a priori, K_a2, K _f , and K_1e could be identifiable. Alternatively, if K_1e can be assumed to be zero, then the remaining parameters K _f , K_a2, and V₂ will be identifiable.

Our previous studies in rats showed that the unchanged quercetin in the systemic circulation is primarily metabolized to glucuronide/sulfate conjugates, and only small amount of quercetin (less than 1%) was excreted into the urine and bile. Thus it is reasonable to assume that the excretion pathway of the unchanged quercetin be ignored (i.e., K₁₀ = 0). With this assumption, the proposed model is globally identifiable and all the parameters can be identifiable when both iv and oral doses are administered.

Integrated Solutions for the Plasma Concentrations of Unchanged Quercetin and Glucuronide/Sulfate Conjugates

With an assumption of K₁₀ = 0, solving Eqs.  (A1)–(A4) with initial conditions (at time t = 0, X₁₍₀₎ = D, X₂₍₀₎ = X₃₍₀₎ = X₄₍₀₎ = 0), the plasma concentration of unchanged quercetin in compartment 1, C₁(t), and its glucuronide/sulfate conjugates in compartment 3, C₃(t), at any time after an iv administration can be expressed by (Eqs. (A19) and (A20)):

$$ C_{1} {\left( t \right)} = \frac{D} {{V_{1} }} \cdot \left[ {\frac{{k_{{21}} - \alpha _{1} }} {{\beta _{1} - \alpha _{1} }}e^{{ - \alpha _{1} t}} } \right. + \left. {\frac{{K_{{21}} - \beta _{1} }} {{\alpha _{1} - \beta _{1} }}e^{{ - \beta _{1} t}} } \right] $$

(A19)

$$ C_{3} {\left( t \right)} = \frac{{DK_{f} }} {{V_{3} }} \cdot {\left[ {\lambda _{1} e^{{ - \alpha _{1} t}} + \lambda _{2} e^{{ - \beta _{1} t}} + \lambda _{3} e^{{ - \alpha _{2} t}} + \lambda _{4} e^{{ - \beta _{2} t}} } \right]} $$

(A20)

$$ where\;\lambda _{1} = \frac{{{\left( {K_{{21}} - \alpha _{1} } \right)}{\left( {K_{{43}} - \alpha _{1} } \right)}}} {{{\left( {\beta _{1} - \alpha _{1} } \right)}{\left( {\alpha _{2} - \alpha _{1} } \right)}{\left( {\beta _{2} - \alpha _{1} } \right)}}} $$

(A21)

where

$$ \lambda _{2} = \frac{{{\left( {K_{{21}} - \beta _{1} } \right)}{\left( {K_{{43}} - \beta _{1} } \right)}}} {{{\left( {\alpha _{1} - \beta _{1} } \right)}{\left( {\alpha _{2} - \beta _{1} } \right)}{\left( {\beta _{2} - \beta _{1} } \right)}}} $$

(A22)

$$ \lambda _{3} = \frac{{{\left( {K_{{21}} - \alpha _{2} } \right)}{\left( {K_{{43}} - \alpha _{2} } \right)}}} {{{\left( {\alpha _{1} - \alpha _{2} } \right)}{\left( {\beta _{1} - \alpha _{2} } \right)}{\left( {\beta _{2} - \alpha _{2} } \right)}}} $$

(A23)

$$ \lambda _{4} = \frac{{{\left( {K_{{21}} - \beta _{2} } \right)}{\left( {K_{{43}} - \beta _{2} } \right)}}} {{{\left( {\alpha _{1} - \beta _{2} } \right)}{\left( {\beta _{1} - \beta _{2} } \right)}{\left( {\alpha _{2} - \beta _{2} } \right)}}} $$

(A24)

$$ \alpha _{1} + \beta _{1} = K_{f} + K_{{12}} + K_{{21}} $$

(A25)

$$ \alpha _{1} \beta _{1} = K_{f} K_{{21}} $$

(A26)

$$ \alpha _{2} + \beta _{2} = K_{{30}} + K_{{34}} + K_{{43}} $$

(A27)

$$ \alpha _{2} \beta _{2} = K_{{30}} K_{{43}} $$

(A28)

In the above Eqs. (A19)–(A28), D is the dose given intravenously, t is the time after iv administration, α₁ and β₁ are the first-order hybrid rate constants for the distribution and elimination phases for unchanged quercetin, respectively; whereas α₂ and β₂ are the first-order hybrid rate constants for the distribution and elimination phases for glucuronide/sulfate conjugates, respectively.

The integrated solutions for plasma concentrations of unchanged quercetin (in compartment 1) and its glucuronide/sulfate conjugates (in compartment 3) after an oral administration are:

$$ C_{1} {\left( t \right)} = \frac{{F_{1} DK_{{a1}} }} {{V_{1} }} \cdot {\left[ {\frac{{k_{{21}} - K_{{a1}} }} {{{\left( {\alpha _{1} - K_{{a1}} } \right)}{\left( {\beta _{1} - K_{{a1}} } \right)}}}e^{{ - K_{{a1}} t}} + \frac{{K_{{21}} - \alpha _{1} }} {{{\left( {K_{{a1}} - \alpha _{1} } \right)}{\left( {\beta _{1} - \alpha _{1} } \right)}}}e^{{ - \alpha _{1} t}} + \frac{{K_{{21}} - \beta _{1} }} {{{\left( {K_{{a1}} - \beta _{1} } \right)}{\left( {\alpha _{1} - \beta _{1} } \right)}}}e^{{ - \beta _{1} t}} } \right]} $$

(A29)

$$ C_{3} {\left( t \right)} = \frac{{F_{1} DK_{{a1}} K_{f} }} {{V_{3} }} \cdot {\left[ {A_{1} e^{{ - K_{{a1}} t}} + A_{2} e^{{ - \alpha _{1} t}} + A_{3} e^{{ - \beta _{1} t}} + A_{4} e^{{ - \alpha _{2} t}} + A_{5} e^{{ - \beta _{2} t}} } \right]} + \frac{{F_{2} DK_{{a2}} }} {{V_{3} }} \cdot {\left[ {B_{1} e^{{ - K_{{a2}} t}} + B_{2} e^{{ - \alpha _{2} t}} + B_{3} e^{{ - \beta _{2} t}} } \right]} $$

(A30)

where

$$ A_{1} = \frac{{{\left( {K_{{21}} - K_{{a1}} } \right)}{\left( {K_{{43}} - K_{{a1}} } \right)}}} {{{\left( {\alpha _{1} - K_{{a1}} } \right)}{\left( {\beta _{1} - K_{{a1}} } \right)}{\left( {\alpha _{2} - K_{{a1}} } \right)}{\left( {\beta _{2} - K_{{a1}} } \right)}}} $$

(A31)

$$ A_{2} = \frac{{{\left( {K_{{21}} - \alpha _{1} } \right)}{\left( {K_{{43}} - \alpha _{1} } \right)}}} {{{\left( {K_{{a1}} - \alpha _{1} } \right)}{\left( {\beta _{1} - \alpha _{1} } \right)}{\left( {\alpha _{2} - \alpha _{1} } \right)}{\left( {\beta _{2} - \alpha _{1} } \right)}}} $$

(A32)

$$ A_{3} = \frac{{{\left( {K_{{21}} - \beta _{1} } \right)}{\left( {K_{{43}} - \beta _{1} } \right)}}} {{{\left( {K_{{a1}} - \beta _{1} } \right)}{\left( {\alpha _{1} - \beta _{1} } \right)}{\left( {\alpha _{1} - \beta _{1} } \right)}{\left( {\beta _{2} - \beta _{1} } \right)}}} $$

(A33)

$$ A_{4} = \frac{{{\left( {K_{{21}} - \alpha _{2} } \right)}{\left( {K_{{43}} - \alpha _{2} } \right)}}} {{{\left( {K_{{a1}} - \alpha _{2} } \right)}{\left( {\alpha _{1} - \alpha _{2} } \right)}{\left( {\beta _{1} - \alpha _{2} } \right)}{\left( {\beta _{2} - \alpha _{2} } \right)}}} $$

(A34)

$$ A_{5} = \frac{{{\left( {K_{{21}} - \beta _{2} } \right)}{\left( {K_{{43}} - \beta _{2} } \right)}}} {{{\left( {K_{{a1}} - \beta _{2} } \right)}{\left( {\alpha _{1} - \beta _{2} } \right)}{\left( {\beta _{1} - \beta _{2} } \right)}{\left( {\alpha _{2} - \beta _{2} } \right)}}} $$

(A35)

$$ B_{1} = \frac{{K_{{43}} - K_{{a2}} }} {{{\left( {\alpha _{2} - K_{{a2}} } \right)}{\left( {\beta _{2} - K_{{a2}} } \right)}}} $$

(A36)

$$ B_{2} = \frac{{K_{{43}} - \alpha _{2} }} {{{\left( {K_{{a2}} - \alpha _{2} } \right)}{\left( {\beta _{2} - \alpha _{2} } \right)}}} $$

(A37)

$$ B_{3} = \frac{{K_{{43}} - \beta _{2} }} {{{\left( {K_{{a2}} - \beta _{2} } \right)}{\left( {\alpha _{2} - \beta _{2} } \right)}}} $$

(A38)