Compartment Model

Pharmacokinetic Models and Equations

Pharmacokinetic models are useful to describe data sets, to predict serum concentrations after several doses or different routes of administration, and to calculate pharmacokinetic constants such as CL, V_D, andt_1/2.¹¹ Compartmental models depict the body as one or more discrete compartments to which a drug is distributed and/or from which a drug is eliminated. The shape of the serum-concentration-versus-time curve determines the number of compartments in the pharmacokinetic model and the equation used in computations . First-order rate constants, known as microconstants, describe the rate of transfer from one compartment to another. Each compartment also has its own V_D. For clinical dosage adjustment purposes using drug concentrations, a one-compartment model is the most commonly used pharmacokinetic model.

Visual representations of one- and two-compartment drug-distribution models One-Compartment Model

the simplest case uses a single compartment to represent the entire body . The drug enters the compartment by continuous IV infusion (k₀), absorption from an extravascular site with an absorption rate constant of k_a, or IV bolus (D). After an IV bolus, serum concentrations decline in a straight line when plotted on semilogarithmic coordinates . The slope of the line is –k/2.303; t_1/2 can be computed by determining the time required for concentrations to decrease by one-half (t_1/2 = 0.693/k). The equation that describes the data is C = (D/V_D)e^–kt. V_D is calculated by dividing the IV dose by the y intercept (the concentration at time zero, C₀) of the graph. CL is computed by taking the product of k and V_D. Once V_D and kare known, concentrations at any time after the dose can be computed [C = (D/V_D)e^–kt].

When an extravascular dose is given, one-compartment-model serum concentrations rise during absorption, reach C_max, then decrease in a straight line with a slope equal to –k/2.303. The equation that describes the data is C = {(FDk_a)/[V_D(k_a – k)]}(e^–kt – e^–k_at), where F is the fraction of the dose absorbed into the systemic circulation. The absorption rate constant (k_a) is obtained using the method of residuals.

The method of residuals is used to obtain the individual rate constants . A is determined by extrapolating the terminal slope to the y axis; k can be obtained by calculating the slope or t_1/2 and using the formulas given for the IV bolus case. At each time point in the absorption portion of the curve, the concentration value from the extrapolated line is noted and called the extrapolated concentration. For each point, the actual concentration is subtracted from the extrapolated concentration to compute the residual concentration. When the residual concentrations are plotted on semilogarithmic coordinates , inset), a line with y intercept equal to A and slope equal to –k_a/2.303 is obtained. When these values are calculated, they can be placed into the equation (C = Ae^–kt – Ae^–k_at, where A = FDk_a/[V_D(k_a – k)]) and used to compute the serum concentration at any time after the extravascular dose. The intercepts and rate constants also can be used to compute CL and V_D: CL = FD/(A/k – A/k_a) and V_D = CL/k, where F is the fraction of the dose absorbed into the systemic circulation.

Calculation of the half-life of a drug following oral, intramuscular, or other extravascular dosing route. Inset shows calculation of the absorption rate constant (ka) using the method of residuals.

During a continuous IV infusion, the serum concentrations in a one-compartment model change according to the following function: C = (k₀/CL)(1 – e^–kt). If the infusion has been running for more than three to five half-lives, the patient will be at steady state, and CL can be calculated (CL = k₀/C_ss). When the infusion is discontinued, serum concentrations appear to decline in a straight line when plotted on semilogarithmic paper with a slope of –k/2.303. V_D is computed by dividing CL by k

Achievement of steady-state serum concentrations after three to five half-lives of a drug. Note the elimination phase after discontinuance of the infusion.

Multicompartment Model

After an IV bolus dose, serum concentrations often decline in two or more phases. During the early phases, the drug leaves the bloodstream by two mechanisms: (a) distribution into tissues and (b) metabolism and/or elimination. Because the drug is leaving the bloodstream through these two mechanisms, serum concentrations decline rapidly. After tissues and blood are in equilibrium, only metabolism and elimination remove the drug from the blood. During this terminal phase, serum concentrations decline more slowly. Thehalf-life is measured during the terminal phase by determining the time required for concentrations to decline by one-half.

After an IV bolus dose, serum concentrations decrease as if the drug were being injected into a central compartment that not only metabolizes and eliminates the drug but also distributes the drug to one or more other compartments. Of these multicompartment models, the two-compartment model is encountered most commonly . After an IV bolus injection, serum concentrations decrease in two distinct phases, described by the equation

or C = Ae^–t + Be^–t, where k₂₁ is the first-order rate constant that reflects the transfer of the drug from compartment 2 to compartment 1, V_D1 is the V_D of compartment 1, A = D( – k₂₁)/[V_D1( – )] and B = D(k₂₁ –)/[V_D1( – )]. The rate constants  and  found in the exponents of the equations describe the distribution and elimination of the drug, respectively . A and B are the y intercepts of the lines that describe drug distribution and elimination, respectively, on the log concentration-versus-time plot.

Calculation of  and  half-lives following IV dosing. Inset shows calculation of the distribution rate constant () using the method of residuals.

The residual line is calculated as before using the method of residuals , inset). The terminal line is extrapolated to the y axis, and extrapolated concentrations are determined for each time point. Because actual concentrations are greater in this case, residual concentrations are calculated by subtracting the extrapolated concentrations from the actual concentrations. When plotted on semilogarithmic paper, the residual line has a y intercept equal to A. The slope of the residual line is used to compute  (slope = –/2.303). With the rate constants ( and ) and the intercepts (A and B), concentrations can be calculated for any time after the IV bolus dose (C = Ae^–t + Be^–t), or pharmacokinetic constants can be computed: CL =D/[(A/) + (B/)], V_D, = CL/, V_D,ss = {D[(A/²) + (B/²)]}/[(A/) + (B/)]².

If serum concentrations of a drug given as a continuous IV infusion decline in a biphasic manner after the infusion is discontinued, a two-compartment model describes the data set  In this instance, the postinfusion concentrations decrease according to the equation C = Re^–t' + Se^–t', where t' is the postinfusion time (t' = 0 when infusion is discontinued), and R, S, , and  are determined from the postinfusion concentrations using the method of residuals with the y axis set at t' = 0. R and S are used to compute A and B. A and B are the y intercepts that would have occurred had the total dose given during the infusion (D = k₀T) been administered as an IV bolus dose:

Calculation of  and  half-lives following a steady-state infusion. Inset shows calculation of the distribution rate constant () using the method of residuals.

where T is the duration of infusion. Once A, B, , and  are known, the equations for an IV bolus are used to compute the pharmacokinetic constants. Often, when a drug is given as an IV bolus or continuous IV infusion, a two-compartment model is used to describe the data, but when the same agent is given extravascularly, a one-compartment model applies.¹⁴ In this case, distribution occurs during the absorption phase, so a distribution phase is not observed.

Volumes of Distribution in Multicompartment Models

Two different V_D values are needed as proportionality constants for drugs that require multicompartment models to describe the serum-concentration-versus-time curve. The V_D that is used to compute the amount of drug in the body during the terminal () portion of the curve is called V_D,  (amount of drug in body = V_D, C). During a continuous IV infusion at steady state, V_D,ss is used to compute the amount of drug in the body (amount of drug in body = V_D,ssC). V_D,ss is also the V_D that can be computed using the physiologic volumes of blood and tissues and the ratio of unbound drug in blood to that in tissues [V_D,ss = V_b + (f_b/f_t)V_t]. Because the value of V_D, changes when CL changes, V_D,ss should be used to indicate if drug distribution changes during pharmacokinetic or drug-interaction experiments.

Multiple Dosing and Steady-State Equations

Any of these compartmental equations can be used to determine serum concentrations after multiple doses. The multiple-dosing factor (1 – e^–nK)/(1 – e^–K), where n is the number of doses, K is the appropriate rate constant, and  is the dosage interval, is simply multiplied by each exponential term in the equation, substituting the rate constant of each exponent for K. Time (t) is set at 0 at the beginning of each dosage interval. For example, a single-dose two-compartment IV bolus is calculated as follows: C = Ae^–t + Be^–t. Thus, the equation for a multiple-dose two-compartment IV bolus is

A single-dose one-compartment IV bolus is calculated as C = (D/V_D)e^–kt. For a multiple-dose one-compartment IV bolus, the concentration is C = (D/V_D)e^–kt[(1 – e^–nk)/(1 – e^–k)].

At steady state, the number of doses becomes large, e^–nK approaches zero, and the multiple-dosing factor equals 1/(1 – e^–K). Therefore, the steady-state versions of the equations are simpler than their multiple-dose counterparts:

and

for a steady-state two-compartment IV bolus and a steady-state one-compartment IV bolus, respectively.