Physiologically Based Pharmacokinetic (PBPK) Modeling and Simulations. Sheila Annie Peters

only elimination pathway for the parent drug and a complete release of the metabolite into systemic circulation without prior elimination in the organ where it is formed is assumed.

       1.2.10.4 Limiting Conditions

Equation 1.50 suggests three different possibilities:

      1 Metabolite disposition is formation rate‐limited (kel,M > kf,M ) – commonly observed

      2 Metabolite disposition is elimination rate‐limited (kf,M > kel,M )

      3 Neither of the two steps above are rate limiting

The first 2 cases are illustrated in Figure 1.9.

      In the first case, when metabolite disposition is limited by its formation, the half‐life of parent is greater than that of its metabolite and the following equation holds:

(1.51)

      CL_M is the clearance of the metabolite, CL_f,M is the metabolic clearance of the parent drug related to the formation of metabolite, V_M is the volume of distribution of the metabolite, and V is the volume of distribution of the parent drug. The PK profiles of the parent drug and its metabolite are characterized by parallel terminal slopes (see Figure 1.9). The metabolite is eliminated as soon as it is formed. Consequently, the right‐hand side of Equation 1.50 becomes zero leading to

(1.52)

Rearranging Equation 1.52,

      (1.53)

If the parent and metabolite have similar volumes of distribution, according to Equation 1.51, the clearance of the metabolite is greater than that of the parent drug. Thus, the amount of parent drug in the body is always greater than the metabolite. If the volume of distribution of the metabolite is much smaller compared to that of parent drug, then according to Equation 1.51, it is possible that the ratio of the elimination rate constants of the parent and metabolite is ≫1, provided V_M /V is smaller than CL_M /CL_f,M . It is therefore not necessary that CL_M should be greater than CL_f,M . In other words, the amount of parent drug in the body need not be always greater than the metabolite, if the metabolite is polar and has a relatively low volume of distribution compared to the more lipophilic parent drug (see Figure 1.9). An example of this is propranolol where the amount of parent compound is less than that of the metabolite.

      In the second case, the metabolite disposition is limited by its elimination and the half‐life of the metabolite is greater than that of its parent drug. This may lead to accumulation of the metabolite in the body and requires careful safety monitoring.

      Lastly, in the event of comparable half‐lives of drug and metabolite, the terminal slope of the metabolite appears to decline slower than would be expected from its elimination rate constant, sustained by the parent drug.

Figure 1.9. Limiting conditions on metabolite disposition. (a) Formation‐limited metabolite kinetics when the volume of distribution of parent is similar to the volume of distribution of its metabolite. (b) Formation‐limited metabolite kinetics when the volume of distribution of parent is greater than the volume of distribution of its metabolite. (c) Elimination‐limited metabolite kinetics.

       1.2.10.5 Drug‐Metabolite Plasma Concentration Relationships After Single Drug Administration

Detailed characterization of metabolite PK is limited by the availability of pure standards and chemical stability of metabolites. However, when the metabolite is likely to be active or toxic, the plasma concentration‐time profile of the metabolite is measured. Analogous to Equation 1.50, one can obtain Equation 1.54, recognizing that the product of elimination rate constant and amount of drug is the same as the product of clearance and concentration (see Equation 1.2)

(1.54)

      Integrating Equation 1.54,

(1.55)

      where AUC_M and AUC are the areas under the concentration‐time profiles of metabolite and parent drug, respectively. Recognizing that CL_f,M is the product of fraction metabolized (f_m ) and the total clearance (CL) of the parent drug,

(1.56)

Substituting for CL_f,M from Equation 1.56 in Equation 1.55,

      (1.57)

f_m is obtained by measuring the amount of metabolite excreted in urine collected over a sufficient length of time (see Equation 1.29).

       1.2.10.6 Steady‐State Metabolite Kinetics

      At steady state, the rate of change in the amount of metabolite in the body is zero, making the left‐hand sides of Equations 1.50 and 1.54 equal zero. It follows that the amount and concentration of the metabolite in the body at steady state are:

      (1.58)

       1.2.10.7 First‐Pass Metabolite Production

Following oral drug administration,