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

follows:

(1.21)

According to Equation 1.21, considering an average H of 0.45, the minimum value of R can be 0.55 which corresponds to no distribution into erythrocytes. However, there is no upper limit. For tacrolimus, R is as high as 55 and exhibits concentration dependence (Jusko et al. 1995). R can also be predicted (Paixão et al., 2009).

      Partitioning can be fast for some drugs and distribution equilibrium is reached within a few seconds to minutes. However, many drugs with primary amine groups show delayed equilibrium probably due to the formation of Schiff bases with membrane fatty acids and aldehydes. While the displacement of the plasma–protein–bound drug to the unbound is rapid (except for protein molecules), displacement of erythrocyte bound drug is relatively slow. For acids with high plasma protein binding, distribution into erythrocytes can significantly affect its distribution volume, as other tissue compartments are not as significant. If blood–plasma concentration ratios exceed 1, as is the case for lipophilic bases, then plasma clearance significantly overestimates blood clearance and could even exceed hepatic blood flow. This is because the concentrations measured in plasma will always be much smaller compared to that measured in whole blood. This is due to greater distribution into the erythrocytes when R is >1. Thus, blood clearance is related to plasma clearance and blood–plasma ratio by the following equation:

      (1.22a)

      Similarly, plasma and blood volume are related by

      (1.22b)

1.2.4 Hepatic, Renal, and Biliary Clearances

The CL estimated from dose and AUC (Equation 1.10) is the overall clearance of the drug from blood, also called the total clearance. Several clearance pathways could contribute to the total clearance of a drug. Hepatic metabolism, renal, and biliary pathways are some of the clearance routes available for a xenobiotic. The most common route is hepatic metabolism. A drug may utilize one or more clearance pathways depending on its physicochemical properties. The total clearance of a drug is the sum of its hepatic (CL_H ), renal (CL_R ), and biliary (CL_B ) clearances.

      (1.23)

      CL_organ is the clearance from an eliminating organ, Q_organ is the blood flow rate to that organ, C_ART and C_VEN are the arterial and venous concentrations. (Q_organ × C_ART − Q_organ × C_VEN ) is the rate of elimination from that organ.

       1.2.4.1 Hepatic Clearance

      The liver is the most important eliminating organ, where phase I and phase II metabolizing enzymes convert low molecular weight drugs into more hydrophilic compounds with greater molecular weight which can enter bile or undergo renal elimination. About 40 human CYP450 genes have been cloned and classified according to sequence homology. Of these, only 3 CYP450 families and <12 unique enzymes play a substantial role in the hepatic metabolism of drugs in humans. The rate of such an enzyme driven biotransformation reaction, v, depends on the free concentration of the drug, C, according to the Michaelis–Menten equation:

(1.24)

CL_int is the intrinsic clearance of the drug, v_max is the maximum velocity of the reaction, and K_M is the Michaelis–Menten constant (Figure 1.3). When the therapeutic concentration range is low relative to K_M (true for many drugs), Equation 1.24 becomes:

(1.25)

Figure 1.3. Rate of an enzyme‐catalyzed reaction as a function of substrate concentration.

Equation 1.25 suggests that the rate of a metabolic reaction varies linearly with the drug concentration at concentrations not exceeding K_M . CL_int then equals the ratio of v_max to K_M and is independent of the drug concentration. However, when drug concentrations are comparable with or exceeds K_M, which can be the case at high drug doses, CL_int is dependent on C (see Equation 1.24). The greater the drug concentration, the greater the extent of enzyme saturation and smaller the value of CL_int . A maximal rate of metabolism v_max is reached at concentrations much higher relative to K_M . Under these conditions, zero‐order kinetics is said to prevail, and a constant amount of drug is eliminated per unit time, independent of the amount of drug in the body. Well‐known examples of drugs exhibiting nonlinear clearance include phenytoin, ethanol, methyl salicylate, and theophylline (in some individuals). Apart from CL_int, the hepatic clearance CL_H is dependent on how fast the drug is delivered to the enzymes in the hepatocytes determined by the total blood flow rate to the liver, Q_LI . A basic tenet of pharmacokinetics is that only the fraction of drug that is not bound to blood, f_ub, is available for distribution into tissues and for clearance. The dependencies of CL_H on CL_int , Q_LI, and f_ub are encapsulated in the well‐stirred model, in which drug concentration in the liver is assumed to be uniform throughout the tissue. According to this model, the hepatic clearance from blood, CL_H, is given by:

(1.26)

Equation 1.26 shows that when the product f_up × CL_int is high compared to Q_LI, then CL_H is simply determined by the blood flow rate to the liver. The drug clearance is limited by the rate at which it is delivered to the drug metabolizing enzymes. On the other hand, when the product f_up × CL_int is low compared to Q_LI, then the hepatic clearance linearly varies with the product of fraction unbound in plasma and the intrinsic clearance.

      The fraction of drug unbound in blood, f_ub is related to f_up and R as follows:

      (1.27)