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

of the drug relative to its K_M . If the affinity of the drug to the protein is high (low K_M ) and/or the drug concentrations are high, the transporters are easily saturated. For example, P‐gp efflux in the enterocytes is easily saturated by many of its substrates. The rate of P‐gp efflux in the gut also depends on the membrane permeability of the drug and the rate of gut metabolism. For high permeability drugs with high gut extraction, the rate of efflux is low. The impact of uptake or efflux transporters in increasing or decreasing intracellular concentrations of their substrates is much more for low permeability drugs compared to high permeability drugs. P‐gp and CYP3A4 share a common substrate specificity. In the gut, where drug concentrations can be high, this enzyme–transporter interplay (Benet, 2009) increases the probability of contact between the drug and enzyme, by making it possible for CYP3A4 to remain unsaturated due to the dilution effects of P‐gp efflux. The mean residence time of drug and enzyme also increases, as the drug is presented to the enzymes repeatedly while it transits down the small intestine. P‐gp efflux however, also offers a competing mechanism to metabolism both in the liver and in the gut. The roles of renal anionic and cationic transporters in drug disposition have been reviewed (Dresser et al., 2001; Masereeuw and Russel, 2010).

1.2.8 Linear and Non‐Linear Pharmacokinetics

Most drugs are expected to have linear PK at therapeutic concentrations. However, multiple factors can contribute to nonlinearity even at therapeutic concentrations. For drugs in infection and cancer therapeutic areas, where therapeutic doses are generally high, the high drug concentrations can saturate the enzymes and transporters that are involved in their metabolism and elimination in the gut, liver, and other elimination organs. For example, nonlinear PK is frequently observed for antiretroviral drugs. Drugs that inhibit or induce enzymes can cause auto‐inhibition or auto‐induction, leading to time‐ or concentration‐dependent changes in exposure. Drugs binding plasma proteins, especially to AGP can also exhibit nonlinear kinetics due to the saturation of protein binding. Certain drugs (most monoclonal antibody drugs and some small molecules) bind to their pharmacological target (such as a receptor) with high affinity. The internalization of the receptor–drug complex triggered by the high‐affinity receptor binding can then influence the distribution and elimination of the drug. This phenomenon is called target‐mediated drug disposition (TMDD). Saturation of enzyme‐, transporter‐ and plasma proteins, autoinhibition, autoinduction, and TMDD‐driven clearance leads to higher exposures at higher doses, while autoinduction leads to lower exposures at higher doses. Poorly soluble drugs with high therapeutic doses are at a risk of incomplete absorption and therefore lower exposures at higher doses. An important clinical implication of nonlinear pharmacokinetics (Ludden, 1991) is the altered half‐life, which leads to a longer or shorter time to achieve a given fraction of steady state. Since dose predictions are often done for the steady state, nonlinearity makes any predictions of disposition highly uncertain.

1.2.9 Intravenous Infusion, Repeated Dosing, Steady State Kinetics, and Accumulation

      For short half‐life drugs that require plasma or tissue concentrations to be maintained at the therapeutic level for a short treatment period, a constant‐rate IV infusion administered in hospital‐settings via a drip or pump offers the best solution. With a constant‐rate infusion, the rate of change in the amount of drug in plasma is the difference between the rate of drug infusion, R₀, (what goes in) and its rate of elimination (what goes out). Expressing this mathematically (see Equations 1.1 and 1.2),

(1.40)

(1.41)

At steady state (SS), the rates of what goes in and what goes out are equal and there is no net change in the drug amount or concentration in the plasma. In other words, the left‐hand sides of both Equations 1.40 and 1.41 become zero. It follows that at steady state, the amount and the concentration of a drug in plasma at steady state are given by

(1.42)

      (1.43)

Recognizing that MRT is the same as the inverse of the elimination rate constant (i.e., time spent by a molecule in the body), it follows from Equations 1.42 and 1.2 that

      (1.44)

      Thus, knowing MRT and CL, the steady state volume of distribution (V_SS ) of a drug can be estimated.

The elimination of an IV bolus dose generally follows an exponential decay starting from an initial concentration, C₀. If the initial concentration is assumed to be the steady state concentration, C_ss, the concentration of the drug at any time, t, (C_t ), after the cessation of infusion is given by:

      (1.45)

When a drug is infused intravenously at a constant rate, the plasma concentration continues to rise until elimination equals the rate of delivery into the body, at which point a steady state is said to have been reached. This is illustrated in Figure 1.8a. Mathematically, the time dependence of the infusion curve is obtained by subtracting the exponential term from 1 and expressed as follows:

(1.46)

C_t (inf) is the concentration of the drug at any time t, following a constant rate infusion of the drug. Equation 1.46 suggests that this concentration will tend towards the steady state concentration, as t approaches infinity. Also, regardless of the drug, 50% of the plateau concentration is attained in 1 half‐life of the drug. 75, 87.5, and 93.75% of the plateau concentration are reached in 2, 3, and 4 half‐lives, respectively. For all practical purposes, the time to reach steady state is about 3–5 half‐lives. Thus, the time required to reach steady state depends only on the drug’s half‐life. The shorter the half‐life, the more rapidly the steady state is reached. The size of the dose and the route of administration have little effect. Figure 1.8b shows the concentration‐time profile of a successively administered oral drug. The profile parallels that observed for the constant rate infusion. However, fluctuations occur within each dosing interval, as a dose is absorbed and eliminated, leading to a C_ss,max and a C_ss,min . C_av is the average of C_ss,max and C_ss,min . The C_ss after an IV infusion or C_ss,av following repeated oral doses are simply given by the ratio of dosing rate to clearance and given by: