Kinetics of Continuous Administration

The preceding discussion describes the pharmacokinetic processes that determine the rates of absorption, distribution, and elimination of a drug. Pharmacokinetics also describes the quantitative, time-dependent changes of both the plasma drug concentration and the total amount of drug in the body, following the drug's administration by various routes, with the two most common being IV infusion and oral fixed-dose/fixed-time interval regimens (for example, one tablet every 4 hours). The interactions of the processes previously described determine the pharmacokinetics profile of a drug. The significance of identifying the pharmacokinetics of a drug lies not only in defining the factors that influence its levels and persistence in the body, but also in tailoring the therapeutic use of drugs that have a high toxic potential. 

[Note: The following discussion assumes that the administered drug distributes into a single body compartment. In actuality, most drugs equilibrate between two or three compartments and, thus, display complex kinetic behavior. However, the simpler model suffices to demonstrate the concepts.] At steady state, input (rate of infusion) equals output (rate of elimination).

Figure  At steady state, input (rate of infusion) equals output (rate of elimination).

A. Kinetics of IV infusion

With continuous IV infusion, the rate of drug entry into the body is constant. In the majority of cases, the elimination of a drug is first order; that is, a constant fraction of the agent is cleared per unit of time. Therefore, the rate of drug exit from the body increases proportionately as the plasma concentration increases, and at every point in time, it is proportional to the plasma concentration of the drug. 

1. Steady-state drug levels in blood: 

Following the initiation of an IV infusion, the plasma concentration of drug rises until the rate of drug eliminated from the body precisely balances the input rate. Thus, a steady-state is achieved in which the plasma concentration of drug remains constant.

 [Note: The rate of drug elimination from the body = (CLt)(C), where CL = total body clearance and C = the plasma concentration of drug.] Two questions can be asked about achieving the steady-state. First, what is the relationship between the rate of drug infusion and the plasma concentration of drug achieved at the plateau, or steady state? Second, what length of time is required to reach the steady state drug concentration? 

2. Influence of the rate of drug infusion on the steady state:

A steady-state plasma concentration of a drug occurs when the rate of drug elimination is equal to the rate of administration as described by the following equation: where Css = the steady-state concentration of the drug, Ro = the infusion rate (for example, mg/min), ke is the firstorder elimination rate constant, and Vd = the volume of distribution. Because ke, CLt, and Vd are constant for most drugs showing first-order kinetics, Css is directly proportional to Ro; that is, the steady-state plasma concentration is directly proportional to the infusion rate. For example, if the infusion rate is doubled, the plasma concentration ultimately achieved at the steady state is doubled 

Figure : Effect of infusion rate on the steady-state concentration of drug in the plasma. (Ro = rate of infusion of a drug.)

Furthermore, the steady-state concentration is inversely proportional to the clearance of the drug, CLt. Thus, any factor that decreases clearance, such as liver or kidney disease, increases the steady-state concentration of an infused drug (assuming Vd remains constant). Factors that increase clearance of a drug, such as increased metabolism, decrease the steady-state concentrations of an infused drug

3. Time required to reach the steady-state drug concentration:

The concentration of drug rises from zero at the start of the infusion to its ultimate steady-state level, Css 

Figure : Rate of attainment of steady-state concentration of a drug in the plasma.

The fractional rate of approach to a steady state is achieved by a first-order process. 

a. Exponential approach to steady state:   The rate constant for attainment of steady state is the rate constant for total body elimination of the drug, ke. Thus, fifty percent of the final steady-state concentration of drug is observed after the time elapsed since the infusion, t, is equal to t1/2, where t1/2 (or half-life) is the time required for the drug concentration to change by fifty percent. Waiting another half-life allows the drug concentration to approach 75 percent of Css .The drug concentration is ninety percent of the final steady-state concentration in 3.3 times t1/2. For convenience, therefore, one can assume that a drug will reach steady-state in about four halflives. The time required to reach a specific fraction of the steady-state is described by where f = the fractional shift (for example, 0.9 if the time to reach ninety percent of the steady-state concentration was being calculated) and t = the time elapsed since the start of the infusion. 

b. Effect of the rate of drug infusion:   The sole determinant of the rate that a drug approaches steady state is the t1/2 or ke, and this rate is influenced only by the factors that affect the half-life. The rate of approach to steady state is not affected by the rate of drug infusion. Although increasing the rate of infusion of a drug increases the rate at which any given concentration of drug in the plasma is achieved, it does not influence the time required to reach the ultimate steady-state concentration. This is because the steady-state concentration of drug rises directly with the infusion rate

c. Rate of drug decline when the infusion is stopped:   When the infusion is stopped, the plasma concentration of a drug declines (washes out) to zero with the same time course observed in approaching the steady state This relationship is expressed as where Ct i = the plasma concentration at any time, C0 = the starting plasma concentration, ke = the first-order elimination rate constant, and t = the time elapsed

d. Loading dose:   A delay in achieving the desired plasma levels of drug may be clinically unacceptable. Therefore, a loading  dose of drug can be injected as a single dose to achieve the desired plasma level rapidly, followed by an infusion to maintain the steady state (maintenance dose). In general, the loading dose can be calculated as :

B. Kinetics of fixed-dose/fixed-time-interval regimens

Administration of a drug by fixed doses rather than by continuous infusion is often more convenient. However, fixed doses, given at fixed-time intervals, result in time-dependent fluctuations in the circulating level of drug.

 1. Single IV injection: For simplicity, assume the injected drug rapidly distributes into a single compartment. Because the rate of elimination is usually first order in regard to drug concentration, the circulating level of drug decreases exponentially with time 

Figure : Effect of the dose of a single intravenous injection of drug on plasma levels

[Note: The t1/2 does not depend on the dose of drug administered.] 2. Multiple IV injections: When a drug is given repeatedly at regular intervals, the plasma concentration increases until a steady state is reached

Figure 1.23 Predicted plasma concentrations of a drug given by infusion (A), twice-daily injection (B), or oncedaily injection (C). Model assumes rapid mixing in a single body compartment and a half-life of twelve hours.

Because most drugs are given at intervals shorter than five half-lives and are eliminated exponentially with time, some drug from the first dose remains in the body at the time that the second dose is administered, and some from the second dose remains at the time that the third dose is given, and so forth. Therefore, the drug accumulates until, within the dosing interval, the rate of drug loss (driven by an elevated plasma concentration) exactly balances the rate of drug administration that is, until a steady state is achieved

a. Effect of dosing frequency: The plasma concentration of a drug oscillates about a mean. Using smaller doses at shorter intervals reduces the amplitude of the swings in drug concentration. However, the steady-state concentration of the drug, and the rate at which the steady-state is approached, are not affected by the frequency of dosing.

 b. Example of achievement of steady state using different dosage regimens:  Curve B of Figure 1.23 shows the amount of drug in the body when 1 g of drug is administered IV to a patient and the dose is repeated at a time interval that corresponds to the half-life of the drug. At the end of the first dosing interval, 0.50 units of drug remain from the first dose when the second dose is administered. At the end of the second dosing interval, 0.75 units are present when the third dose is taken. The minimal amount of drug during the dosing interval progressively increases and approaches a value of 1.00 unit, whereas the maximal value immediately following drug administration progressively approaches 2.00 units. Therefore, at the steady state, 1.00 unit of drug is lost during the dosing interval, which is exactly matched by the rate at which the drug is administered that is, the rate  in equals the rate out. As in the case for IV infusion, ninety percent of the steady-state value is achieved in 3.3 times t1/2.

3. Orally administered drugs:  Most drugs that are administered on an outpatient basis are taken orally on a fixeddose/fixed-time-interval regimen for example, a specific dose taken one, two, or three times daily. In contrast to IV injection, orally administered drugs may be absorbed slowly, and the plasma concentration of the drug is influenced by both the rate of absorption and the rate of drug elimination

Figure :Predicted plasma concentrations of a drug given by repeated oral administrations.

This relationship can be expressed as: 

where D = the dose, F = the fraction absorbed (bioavailability),T = dosage interval, Css = the steady-state concentration of the drug, ke = the first-order rate constant for drug elimination from the total body, and Vd = the volume of distribution

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