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Area under the curve and bioavailability
Pharmacological Research, Vol. 42, No. 6, 2000 doi:10.1006/phrs.2000.0719, available online at http://www.idealibrary.com on
AREA UNDER THE CURVE AND BIOAVAILABILITY ALDO RESCIGNO∗ 14767 Square Lake Trail N, Stillwater, MN 55082-9278, USA Accepted 28 June 2000
The Area Under the Curve (AUC) is proportional to the fraction absorbed only if the clearance is constant and the concentration uniform; in all other cases the bioavailability cannot be determined c 2000 Academic Press by comparing AUCs.
K EY WORDS : AUC, clearance, bioavailability, elimination rate.
1
INTRODUCTION
where Cl is clearance, F fraction absorbed, D dose, and AU C the integral from 0 to ∞ of the concentration of a drug in the plasma. This formula is valid even if the drug is not administered as a bolus, but two conditions must be met [1]. First, the clearance must remain constant when the concentration changes; second, the concentration must be uniform throughout the site of elimination. When those two conditions are not valid, formula (1) may be inaccurate, with possibly a very large error [2].
0.8 Concentration
One of the most frequently used formulae in pharmacokinetics is F·D Cl = , (1) AU C
0.6
0.4
0.2
0
1.9
3.9
5.9
7.9 9.9 Time (h)
11.9 13.9
Fig. 1. Plasma concentration of a drug eliminated by a second-order reaction (continuous line: intra-vascular administration, broken line: extra-vascular administration).
DEVELOPMENT Formula (1) is used extensively in the determination of bioavailability. If a drug is administered directly into the plasma as a bolus, the fraction absorbed is obviously equal to 1, therefore from (1) we get D1 Cl1 = , AU C1
(2)
where the subscript 1 refers to the data of this experiment. If we repeat the experiment with the same drug, but administered extra-vascularly, we get Cl2 =
F2 · D2 , AU C2
(3)
where the subscript 2 refers to the data of the second experiment. ∗ E-mail: [email protected]
1043–6618/00/120539–02/$35.00/0
In both experiments we used the same drug, therefore Cl1 = Cl2 and consequently from (2) and (3), D1 F2 · D2 = , AU C1 AU C2 thence
D1 AU C2 . D2 AU C1 What happens when one of the conditions for the validity of formula (1) fails to hold? As an example, consider the case of second-order elimination, i.e. elimination according to equation F2 =
dc = −k · c2 (t); dt
(4)
c 2000 Academic Press
540
Pharmacological Research, Vol. 42, No. 6, 2000
3.5 3
AUC
2.5 2 1.5 1 0.5 0
1.9
3.9
5.9
7.9 9.9 Time (h)
11.9 13.9
Fig. 2. AUC from 0 to t of a drug eliminated by a second-order reaction (continuous line: intra-vascular administration, broken line: extra-vascular administration).
in this case the clearance is obviously not constant, but is a decreasing function of the concentration. If the drug is administered directly into the plasma as a bolus at time t = 0, by integration we get c(t) =
c(0) , 1 + kc(0) · t
thence Z
l
Now with the same dose and extra-vascular administration, dc = −k · c2 (t) + r (t), (5) dt where r (t) is the rate of absorption into the plasma. The integral of this differential equation depends on the absorption function r (t). Figure 1 shows function c(t) from equation (4) and from equation (5) assuming the same elimination rate k and a first-order absorption with unit rate. Figure 2 shows the integrals of the same functions from 0 to t. For t sufficiently large, the estimated AU C of one curve is about double that of the other, even though in both cases the (non-linear) elimination rates, and therefore the clearances, are identical. CONCLUSION The Area Under the Curve is proportional to the fraction absorbed only when the clearance is constant and the concentration uniform; if these two conditions are not met, the determination of bioavailability using the Area Under the Curve may be flawed. In the determination of absolute bioavailability, the error due to a non-constant clearance may be very large due to the fact that the two concentration curves (the Intra Vascular curve and the Extra Vascular curve) are quite different and correspond to possibly different clearances. In the determination of relative bioavailability the error may be smaller if the two concentration curves are not very dissimilar, i.e. if the relative bioavailability is close to one.
c(t)dt = 1/k · ln[1 + kc(0) · t].
0
Observe that this integral does not converge for t → ∞, i.e. the theoretical value of AU C is infinitely large, even though c(t) → 0 when t → ∞; but in practice we estimate the integral from 0 to a finite value of t, and this estimate is always finite.
REFERENCES 1. Rescigno A. Clearance, turnover time, and volume of distribution. Pharmacol Res 1997; 35: 189–93. 2. Rescigno A, Marzo A. Area under the curve, bioavailability, and clearance. J Pharmacokin Biopharm 1991; 19: 473–82.
AREA UNDER THE CURVE AND BIOAVAILABILITY ALDO RESCIGNO∗ 14767 Square Lake Trail N, Stillwater, MN 55082-9278, USA Accepted 28 June 2000
The Area Under the Curve (AUC) is proportional to the fraction absorbed only if the clearance is constant and the concentration uniform; in all other cases the bioavailability cannot be determined c 2000 Academic Press by comparing AUCs.
K EY WORDS : AUC, clearance, bioavailability, elimination rate.
1
INTRODUCTION
where Cl is clearance, F fraction absorbed, D dose, and AU C the integral from 0 to ∞ of the concentration of a drug in the plasma. This formula is valid even if the drug is not administered as a bolus, but two conditions must be met [1]. First, the clearance must remain constant when the concentration changes; second, the concentration must be uniform throughout the site of elimination. When those two conditions are not valid, formula (1) may be inaccurate, with possibly a very large error [2].
0.8 Concentration
One of the most frequently used formulae in pharmacokinetics is F·D Cl = , (1) AU C
0.6
0.4
0.2
0
1.9
3.9
5.9
7.9 9.9 Time (h)
11.9 13.9
Fig. 1. Plasma concentration of a drug eliminated by a second-order reaction (continuous line: intra-vascular administration, broken line: extra-vascular administration).
DEVELOPMENT Formula (1) is used extensively in the determination of bioavailability. If a drug is administered directly into the plasma as a bolus, the fraction absorbed is obviously equal to 1, therefore from (1) we get D1 Cl1 = , AU C1
(2)
where the subscript 1 refers to the data of this experiment. If we repeat the experiment with the same drug, but administered extra-vascularly, we get Cl2 =
F2 · D2 , AU C2
(3)
where the subscript 2 refers to the data of the second experiment. ∗ E-mail: [email protected]
1043–6618/00/120539–02/$35.00/0
In both experiments we used the same drug, therefore Cl1 = Cl2 and consequently from (2) and (3), D1 F2 · D2 = , AU C1 AU C2 thence
D1 AU C2 . D2 AU C1 What happens when one of the conditions for the validity of formula (1) fails to hold? As an example, consider the case of second-order elimination, i.e. elimination according to equation F2 =
dc = −k · c2 (t); dt
(4)
c 2000 Academic Press
540
Pharmacological Research, Vol. 42, No. 6, 2000
3.5 3
AUC
2.5 2 1.5 1 0.5 0
1.9
3.9
5.9
7.9 9.9 Time (h)
11.9 13.9
Fig. 2. AUC from 0 to t of a drug eliminated by a second-order reaction (continuous line: intra-vascular administration, broken line: extra-vascular administration).
in this case the clearance is obviously not constant, but is a decreasing function of the concentration. If the drug is administered directly into the plasma as a bolus at time t = 0, by integration we get c(t) =
c(0) , 1 + kc(0) · t
thence Z
l
Now with the same dose and extra-vascular administration, dc = −k · c2 (t) + r (t), (5) dt where r (t) is the rate of absorption into the plasma. The integral of this differential equation depends on the absorption function r (t). Figure 1 shows function c(t) from equation (4) and from equation (5) assuming the same elimination rate k and a first-order absorption with unit rate. Figure 2 shows the integrals of the same functions from 0 to t. For t sufficiently large, the estimated AU C of one curve is about double that of the other, even though in both cases the (non-linear) elimination rates, and therefore the clearances, are identical. CONCLUSION The Area Under the Curve is proportional to the fraction absorbed only when the clearance is constant and the concentration uniform; if these two conditions are not met, the determination of bioavailability using the Area Under the Curve may be flawed. In the determination of absolute bioavailability, the error due to a non-constant clearance may be very large due to the fact that the two concentration curves (the Intra Vascular curve and the Extra Vascular curve) are quite different and correspond to possibly different clearances. In the determination of relative bioavailability the error may be smaller if the two concentration curves are not very dissimilar, i.e. if the relative bioavailability is close to one.
c(t)dt = 1/k · ln[1 + kc(0) · t].
0
Observe that this integral does not converge for t → ∞, i.e. the theoretical value of AU C is infinitely large, even though c(t) → 0 when t → ∞; but in practice we estimate the integral from 0 to a finite value of t, and this estimate is always finite.
REFERENCES 1. Rescigno A. Clearance, turnover time, and volume of distribution. Pharmacol Res 1997; 35: 189–93. 2. Rescigno A, Marzo A. Area under the curve, bioavailability, and clearance. J Pharmacokin Biopharm 1991; 19: 473–82.