Читать книгу Physiologically Based Pharmacokinetic (PBPK) Modeling and Simulations - Sheila Annie Peters - Страница 32

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:

(1.47)

F is the bioavailability of a non‐IV drug, and τ is the dosing interval, which is 24 hours for a once daily drug. C_ss,av.u is the unbound steady state drug concentration and f_up is the fraction unbound in plasma. According to Equation 1.47, the steady state concentration of the drug increases as clearance decreases. This is the case for drugs exhibiting nonlinear clearance at therapeutic doses either due to enzyme saturation or autoinhibition. In addition, since clearance decreases with increasing doses, the apparent half‐life increases with increasing dose. Consequently, the time to reach steady state increases with dose.

The magnitude of drug concentrations at steady state compared with that after the first dose is determined by the relationship between dosing interval and the half‐life. The ratio of maximum drug concentration under steady state conditions (C_ss,max ) to the maximum drug concentration after the first dose (C_1,max ) is called the accumulation ratio.

Figure 1.8. Steady state concentrations following (a) constant rate infusion (b) oral drug administration.

(1.48)

AUC_SS,τ is the AUC at steady state for a duration of dosing interval, τ. AUC_1,τ is the AUC during the same duration, following administration of the first dose. Equation 1.48 suggests that a drug with a long half‐life compared with its dosing interval is likely to accumulate. According to Equation 1.48, a half‐life greater than approximately six hours will lead to accumulation for a once‐daily regimen. Thus, dosing intervals should consider the half‐lives of drugs to minimize accumulation ratio and thereby to minimize safety risks. For a once daily chronic administration of a drug, a half‐life of around eight hours guarantees an accumulation ratio close to 1.

Since a drug normally requires at least 3–5 half‐lives to reach steady state, effective plasma levels may be achieved more rapidly by the administration of a single large dose called the loading dose to bring the concentration in plasma quickly to the steady state levels followed by maintenance doses. The loading dose required to achieve the plasma levels present at steady state can be determined from the fraction of drug eliminated during the dosing interval and the maintenance dose.

(1.49)