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Relative Bioavailability: What Reference?
Relative Bioavailability: What Reference? To the Editor: It has been proposed that the “Cumulative Relative Fraction Absorbed” method can be used to calculate the cumulative absorption of drug from various test (T) treatments in individual subjects, relative to the reference (R) treatment in these same subjects.’ This equation was derived from the Wagner-Nelson method for calculating percent absorbed2 and, therefore, is restricted to drugs that exhibit monoexponential disposition profiles. The equation used to calculate cumulative relative fraction absorbed (CRFA) is as follows:
half-life changes. Under these circumstances, half-life correction is not warranted, nor is it valid. A major shortcoming of the current CRFA method is that it adjusts only the test treatment AUC terms on the assumption that the half-life observed for the reference product is absolute and reflects the true clearance of the drug. This is not the premise on which half-life correction is based nor is it justified. When clearance changes the half-life correction is warranted, reference treatment and test treatment(s) must all be corrected. Implicit in this method is the assumption that there is a n underlying common “true” clearance; not that the reference treatment has the “true” clearance. This concept is reflected in eq. 2 where both area terms are corrected for half-life. In contrast, eq. 1 for the calculation of where is the concentration of the test (T)product at time t, CRFA uses the half-life for the reference product to correct KZ is the elimination rate constant for the reference (R) all other treatments. This approach can lead to substantial product, AUC:-t is the area under the concentration time error. curve from time zero to time t for the test product and AUCtOne simplistic case is shown in Table I. These data were is the area under the concentration-time curve from time simulated such that the only differences were the elimination zero to infinite time for the reference product. rate constants that yield half-lives of 3 or 5 h. Looking at The purpose of this communication is to (a) expand upon Table I, it is clear that although the true bioavailability (F) one underlying assumption that is critical to the CRFA was unity for both treatments, the uncorrected bioavailabilmethod, ( b ) show the shortcomings of this method as it is ity for Treatment B was 1.67 compared with Treatment A. currently being used, and (c) propose a more robust method This error is resolved by using half-life correction according for future use. In addition, the more robust method will be to eq. 2. In contrast, however, the use of eq. 1 to calculate extended to include comparison of controlled-release (CR) or CRFA for Treatment B, using Treatment A as the reference, sustained-release (SR)products to reference immediate-reresults in a gross overestimation of the CRFA values and a lease (IR) dosage forms using a simultaneous fitting method. substantial decrease in the apparent absorption rate reflectThe underlying assumption that is paramount to all bioed in an apparent absorption half-life change from 1h to >2 availability-bioequivalency testing is that drug clearance is h. Using Treatment B as the reference to calculate the CRFA constant within a subject or group across treatments or that for Treatment A results in even more bizarre observations differences across treatments can be corrected. This has lead including initial CRFA values that approximate the theoretito controversy pertaining to the use of half-life correction in cal values followed by a decline to the relative availability bioavailability-bioequivalency t e ~ t i n gIt . ~is this same genvalue of 0.60. eral principle that has lead to the more common use of The second simplistic example is shown in Table 11. The complete crossover designs and is the reason for reduced power when incomplete block designs are used i n ~ t e a d . ~ parameters used for simulation were identical with those presented for the first case except that F = 0.6 for Treatment Similarly the critical assumption for the CRFA method is B. In this case the observed relative bioavailabilities are that clearance remains constant across all treatments. However, we have one additional problem that compromises our ability to assess this assumption; the precision of our estiTable I-Simulated Theoretical and Observed CRFA Values from mates of clearance as determined by study design and assay the Original (Eq. 1) and the Modified (Eq. 3) CRFA Equations’ methods. Even if clearance and volume of distribution are Treatment B as Treatment A as constant and therefore half-life can be assumed constant, it is Reference Reference likely that, due to sampling-time imprecision and analytical Ti:e’ Theory variability, we will see differences in our estimates of halfOriginal Modified Original Modified life. The critical issue here is whether it can be inferred that (Eq. 1) (Eq.3) (Eq.1) (Eq. 3) this reflects a change in clearance. Koup and Gibaldi3 have 0 0 0 0 0 0 addressed this topic for bioavailability testing and have 1 0.500 0.525 0.500 0.476 0.500 concluded that if half-life estimates are randomly distributed 0.827 0.750 2 0.750 0.678 0.750 for test and reference treatments, then half-life correction is 3 0.875 1.013 0.875 0.750 0.875 warranted if the variance of the corrected bioavailability (F) 1.136 0.938 0.763 0.938 4 0.938 values is less than for the uncorrected values. The equation 1.291 0.984 0.731 0.984 6 0.984 for this correction is as follows: 1.389 0.996 0.690 0.996 8 0.996 16
1.000
24 1 .ooo X
In cases where the half-life estimates are not randomly distributed across treatments, then protracted absorption of one o r more of the dosage forms may be causing the apparent 0022-3549/86/0900-0921$0 l.OO/O 1986,American Pharmaceutical Association
Q
1.000
1.576
1 .ooo
-1.637 _ _ _ 1.000
1.667
1.000
0.615 1.ooo 0.602 1 .ooo 0.600 1 .ooo
a CRFA represents the cumulative relative fraction absorbed. Simulated parameters were k, = 0.693h-’, F = 1 .O, DoselVolume = 1.O, &,= 0.231 h-’ (Treatment A) and 0.1 386 h-’ (Treatment B) where k, is the absorption rate constant.
Journal of Pharmaceutical Sciences / 921 Vol. 75, No. 9, September 1986
Table lCSlmulated Theoretical and Observed CRFA Values from the Orlglnai (Eq. 1) and the Modlfled (Eq. 3) CRFA Equations. Time.
0 1 2 3 4 6 8 16 24 X
Treatment A as Reference
Treatment B as Reference
0
0 0.833 1.250 1.458 1.563 1.640 1.660 1.667 1.667 1.667
0.300 0.450 0.525 0.563 0.590 0.598 0.600 0.600
0.600
0 0.315 0.496 0.608 0.682 0.775
0.833 0.946 0.982 1.000
0
0.300 0.450 0.525 0.563 0.590 0.598 0.600 0.600
0.600
0 0.793 1.130 1.250 1.272 1.218 1.150 1.025 1.003 1.000
0 0.833 1.250 1.458 1.563 1.640 1.660 1.667 1.667 1.667
a CRFA represents the cumulative relative fraction absorbed. Simulated parameters were k, = 0.693 h-’, DoseNolume = 1.O, F = 1.O, &, = 0.231 h - ’ (Treatment A) and F = 0.6, &, = 0.1386 h -’(Treatment B) where k, is the absorption rate constant.
unity whereas the theoretical values are 1.0 and 0.6 for Treatments A and B, respectively. The CRFA profiles are similar to those in Case 1 except that their absolute magnitudes are different. These two examples clearly show the limitations of the CRFA method as it is currently applied. However, these paradoxial observations can be overcome by using the following half-life correction method:
(3) Its potential use or misuse can be based on the same criteria that Koup and Gibaldi3 outlined for the bioavailability halflife correction. If the half-life estimates are randomly distributed among the test and reference treatments and the variance of the corrected CRFA values is less than for the uncorrected value, then this half-life correction method is warranted since, assuming constant distribution volume, it reflects a change in clearance. The proposed CRFA method distributes the error and changes in clearance across all treatments whereas the original method arbitrarily attributed all of the error and changes in clearance to the test treatments by using the reference treatment half-life for all AUC corrections. Ultimately, the concept that an underlying average clearance is the driving force for bioavailability-bioequivalence testing, can be extended by including simultaneous or deconvolution7-9methods. These procedures can be used to determine the intrinsic absorption characteristics of an oral solution of the drug compared with an intravenous dose, to determine the in vivo release characteristics of solid oral dosage forms compared with an oral solution, or to determine
922 /Journal of Pharmaceutical Sciences Vol. 75, No. 9, September 1986
the relative absorption characteristics of controlled-release dosage form compared with an immediate-release reference dosage form. Implicit in these curve-fitting methods is the assumption that clearance remains constant across all treatments or that a single composite clearance can be used for all treatments; e.g., half-life correction within a given subject or patient is implicit in the fitting procedure. Even in cases such as the testing of controlledlsustained-releasedosage forms where “flip-flop” pharmacokinetics may be operative, these methods will find the common clearance term and will isolate the relative absorption rate constants for the various dosage forms. Specifically, in cases where controlled-release or sustained release dosage forms result in complex release profiles, algorithms can be written that reflect simultaneous first-order and zero-order absorption processes or multiple first-order processes, etc. and will isolate the underlying elimination half-life and therefore clearance. In practice, several polyexponential equation forms may need to be tested and compared to assure that both the intrinsic pharmacokinetic profile and the absorption input functions are clearly delineated before choosing an appropriate equation. In conclusion, certain issues have been addressed that have arisen in current bioavailability testing procedures pertaining to the use of the original CRFA method.’ In addition, a modified CRFA equation form, based on currently accepted half-life correction method for bioavailability testing was presented that distributes error and clearance changes across all treatments. Finally, simultaneous fitting and deconvolution methods were recommended to isolate absorption differences without introducing the error associated with clearance changes among treatments as long as the appropriate polyexponential equation can be determined.
References and Notes 1. Welling, P. G.; Patel, R. B.; Patel, U. R.; Gilles ie, W. R.; Craig,
W. A.; Albert, K. S. J . Phnrrn. Sci. 1982, 71.1859. 2. Wagner, J. G.; Nelson, E. J . Phnrm. Sci. 1963, 52,610. 3. Koup, J. R.; Gibaldi, M. Drug Intell. Clin. Phnrrn. 1980,14,321. 4. Colburn, W. A. J . Phnrm. Sci. 1985, 74,195. 5. Patel, I. H.; Bornemann, L.; Colburn, W. A. J . Phnrrn. Sci. 1985, 74,359. 6. Colburn, W. A. J . Phnrm. Sci. 1981, 70,969. I. Cutler, D. J. J . Phnrmokinet. Bio harm. 1978, 6, 221. 8. Gillespie, W. R.; Veng-Pedersen, l? Bwpharm. Drug. Dispos. 1985, 6, 351. 9. Gillespie, W. R.; Veng-Pedersen, P. J . Phnrmacokinet. Biophnrm. 1985,13,289.
WAYNEA. COLBURN’ PETERG WELLING Warner-LambeNParke-Davis
Pharmaceutical Research
Ann Arbor, MI 48105
Received February 24, 1986. Accepted for publication July 17, 1986.
half-life changes. Under these circumstances, half-life correction is not warranted, nor is it valid. A major shortcoming of the current CRFA method is that it adjusts only the test treatment AUC terms on the assumption that the half-life observed for the reference product is absolute and reflects the true clearance of the drug. This is not the premise on which half-life correction is based nor is it justified. When clearance changes the half-life correction is warranted, reference treatment and test treatment(s) must all be corrected. Implicit in this method is the assumption that there is a n underlying common “true” clearance; not that the reference treatment has the “true” clearance. This concept is reflected in eq. 2 where both area terms are corrected for half-life. In contrast, eq. 1 for the calculation of where is the concentration of the test (T)product at time t, CRFA uses the half-life for the reference product to correct KZ is the elimination rate constant for the reference (R) all other treatments. This approach can lead to substantial product, AUC:-t is the area under the concentration time error. curve from time zero to time t for the test product and AUCtOne simplistic case is shown in Table I. These data were is the area under the concentration-time curve from time simulated such that the only differences were the elimination zero to infinite time for the reference product. rate constants that yield half-lives of 3 or 5 h. Looking at The purpose of this communication is to (a) expand upon Table I, it is clear that although the true bioavailability (F) one underlying assumption that is critical to the CRFA was unity for both treatments, the uncorrected bioavailabilmethod, ( b ) show the shortcomings of this method as it is ity for Treatment B was 1.67 compared with Treatment A. currently being used, and (c) propose a more robust method This error is resolved by using half-life correction according for future use. In addition, the more robust method will be to eq. 2. In contrast, however, the use of eq. 1 to calculate extended to include comparison of controlled-release (CR) or CRFA for Treatment B, using Treatment A as the reference, sustained-release (SR)products to reference immediate-reresults in a gross overestimation of the CRFA values and a lease (IR) dosage forms using a simultaneous fitting method. substantial decrease in the apparent absorption rate reflectThe underlying assumption that is paramount to all bioed in an apparent absorption half-life change from 1h to >2 availability-bioequivalency testing is that drug clearance is h. Using Treatment B as the reference to calculate the CRFA constant within a subject or group across treatments or that for Treatment A results in even more bizarre observations differences across treatments can be corrected. This has lead including initial CRFA values that approximate the theoretito controversy pertaining to the use of half-life correction in cal values followed by a decline to the relative availability bioavailability-bioequivalency t e ~ t i n gIt . ~is this same genvalue of 0.60. eral principle that has lead to the more common use of The second simplistic example is shown in Table 11. The complete crossover designs and is the reason for reduced power when incomplete block designs are used i n ~ t e a d . ~ parameters used for simulation were identical with those presented for the first case except that F = 0.6 for Treatment Similarly the critical assumption for the CRFA method is B. In this case the observed relative bioavailabilities are that clearance remains constant across all treatments. However, we have one additional problem that compromises our ability to assess this assumption; the precision of our estiTable I-Simulated Theoretical and Observed CRFA Values from mates of clearance as determined by study design and assay the Original (Eq. 1) and the Modified (Eq. 3) CRFA Equations’ methods. Even if clearance and volume of distribution are Treatment B as Treatment A as constant and therefore half-life can be assumed constant, it is Reference Reference likely that, due to sampling-time imprecision and analytical Ti:e’ Theory variability, we will see differences in our estimates of halfOriginal Modified Original Modified life. The critical issue here is whether it can be inferred that (Eq. 1) (Eq.3) (Eq.1) (Eq. 3) this reflects a change in clearance. Koup and Gibaldi3 have 0 0 0 0 0 0 addressed this topic for bioavailability testing and have 1 0.500 0.525 0.500 0.476 0.500 concluded that if half-life estimates are randomly distributed 0.827 0.750 2 0.750 0.678 0.750 for test and reference treatments, then half-life correction is 3 0.875 1.013 0.875 0.750 0.875 warranted if the variance of the corrected bioavailability (F) 1.136 0.938 0.763 0.938 4 0.938 values is less than for the uncorrected values. The equation 1.291 0.984 0.731 0.984 6 0.984 for this correction is as follows: 1.389 0.996 0.690 0.996 8 0.996 16
1.000
24 1 .ooo X
In cases where the half-life estimates are not randomly distributed across treatments, then protracted absorption of one o r more of the dosage forms may be causing the apparent 0022-3549/86/0900-0921$0 l.OO/O 1986,American Pharmaceutical Association
Q
1.000
1.576
1 .ooo
-1.637 _ _ _ 1.000
1.667
1.000
0.615 1.ooo 0.602 1 .ooo 0.600 1 .ooo
a CRFA represents the cumulative relative fraction absorbed. Simulated parameters were k, = 0.693h-’, F = 1 .O, DoselVolume = 1.O, &,= 0.231 h-’ (Treatment A) and 0.1 386 h-’ (Treatment B) where k, is the absorption rate constant.
Journal of Pharmaceutical Sciences / 921 Vol. 75, No. 9, September 1986
Table lCSlmulated Theoretical and Observed CRFA Values from the Orlglnai (Eq. 1) and the Modlfled (Eq. 3) CRFA Equations. Time.
0 1 2 3 4 6 8 16 24 X
Treatment A as Reference
Treatment B as Reference
0
0 0.833 1.250 1.458 1.563 1.640 1.660 1.667 1.667 1.667
0.300 0.450 0.525 0.563 0.590 0.598 0.600 0.600
0.600
0 0.315 0.496 0.608 0.682 0.775
0.833 0.946 0.982 1.000
0
0.300 0.450 0.525 0.563 0.590 0.598 0.600 0.600
0.600
0 0.793 1.130 1.250 1.272 1.218 1.150 1.025 1.003 1.000
0 0.833 1.250 1.458 1.563 1.640 1.660 1.667 1.667 1.667
a CRFA represents the cumulative relative fraction absorbed. Simulated parameters were k, = 0.693 h-’, DoseNolume = 1.O, F = 1.O, &, = 0.231 h - ’ (Treatment A) and F = 0.6, &, = 0.1386 h -’(Treatment B) where k, is the absorption rate constant.
unity whereas the theoretical values are 1.0 and 0.6 for Treatments A and B, respectively. The CRFA profiles are similar to those in Case 1 except that their absolute magnitudes are different. These two examples clearly show the limitations of the CRFA method as it is currently applied. However, these paradoxial observations can be overcome by using the following half-life correction method:
(3) Its potential use or misuse can be based on the same criteria that Koup and Gibaldi3 outlined for the bioavailability halflife correction. If the half-life estimates are randomly distributed among the test and reference treatments and the variance of the corrected CRFA values is less than for the uncorrected value, then this half-life correction method is warranted since, assuming constant distribution volume, it reflects a change in clearance. The proposed CRFA method distributes the error and changes in clearance across all treatments whereas the original method arbitrarily attributed all of the error and changes in clearance to the test treatments by using the reference treatment half-life for all AUC corrections. Ultimately, the concept that an underlying average clearance is the driving force for bioavailability-bioequivalence testing, can be extended by including simultaneous or deconvolution7-9methods. These procedures can be used to determine the intrinsic absorption characteristics of an oral solution of the drug compared with an intravenous dose, to determine the in vivo release characteristics of solid oral dosage forms compared with an oral solution, or to determine
922 /Journal of Pharmaceutical Sciences Vol. 75, No. 9, September 1986
the relative absorption characteristics of controlled-release dosage form compared with an immediate-release reference dosage form. Implicit in these curve-fitting methods is the assumption that clearance remains constant across all treatments or that a single composite clearance can be used for all treatments; e.g., half-life correction within a given subject or patient is implicit in the fitting procedure. Even in cases such as the testing of controlledlsustained-releasedosage forms where “flip-flop” pharmacokinetics may be operative, these methods will find the common clearance term and will isolate the relative absorption rate constants for the various dosage forms. Specifically, in cases where controlled-release or sustained release dosage forms result in complex release profiles, algorithms can be written that reflect simultaneous first-order and zero-order absorption processes or multiple first-order processes, etc. and will isolate the underlying elimination half-life and therefore clearance. In practice, several polyexponential equation forms may need to be tested and compared to assure that both the intrinsic pharmacokinetic profile and the absorption input functions are clearly delineated before choosing an appropriate equation. In conclusion, certain issues have been addressed that have arisen in current bioavailability testing procedures pertaining to the use of the original CRFA method.’ In addition, a modified CRFA equation form, based on currently accepted half-life correction method for bioavailability testing was presented that distributes error and clearance changes across all treatments. Finally, simultaneous fitting and deconvolution methods were recommended to isolate absorption differences without introducing the error associated with clearance changes among treatments as long as the appropriate polyexponential equation can be determined.
References and Notes 1. Welling, P. G.; Patel, R. B.; Patel, U. R.; Gilles ie, W. R.; Craig,
W. A.; Albert, K. S. J . Phnrrn. Sci. 1982, 71.1859. 2. Wagner, J. G.; Nelson, E. J . Phnrm. Sci. 1963, 52,610. 3. Koup, J. R.; Gibaldi, M. Drug Intell. Clin. Phnrrn. 1980,14,321. 4. Colburn, W. A. J . Phnrm. Sci. 1985, 74,195. 5. Patel, I. H.; Bornemann, L.; Colburn, W. A. J . Phnrrn. Sci. 1985, 74,359. 6. Colburn, W. A. J . Phnrm. Sci. 1981, 70,969. I. Cutler, D. J. J . Phnrmokinet. Bio harm. 1978, 6, 221. 8. Gillespie, W. R.; Veng-Pedersen, l? Bwpharm. Drug. Dispos. 1985, 6, 351. 9. Gillespie, W. R.; Veng-Pedersen, P. J . Phnrmacokinet. Biophnrm. 1985,13,289.
WAYNEA. COLBURN’ PETERG WELLING Warner-LambeNParke-Davis
Pharmaceutical Research
Ann Arbor, MI 48105
Received February 24, 1986. Accepted for publication July 17, 1986.