LEADING ARTICLE
Clin. Pharmacokinet. 22 (4): 247-253. 1992 0312-5963/ 92/0004-0247/$03.50/0 © Adis International Limited. All rights reserved. CPKI
Controversies in Bioequivalence Studies Volker W. Steinijans, Dieter Hauschke and Jan H.G. Jonkman Department of Biometry, Byk Gulden Pharmaceuticals, Konstanz, Germany, and Institute for Clinical Pharmacology, Pharma Bio-Research Int. B.V., Science Park, Zuidlaren, The Netherlands
The design, performance and evaluation of bioequivalence studies have received major attention from academia, the pharmaceutical industry and health authorities over the past decade. Despite the efforts of various regional and national bodies in setting up guidelines and recommendations (APV 1987; CPMP 1991; FDA 1977, 1988; Nordic Council of Medicines 1987; Skelly 1984), a universal standard is still lacking. Such a standard is becoming ever more necessary, in order to avoid costly and unethical replications of largely identical studies. Although the requests by the various health authorities may appear to differ only marginally, they may lead to different conclusions with regard to approval or rejection of bioequivalence_ Among the issues discussed during a recent workshop of the Drug Information Association (DIA) on 'Bioavailability/Bioequivalence: Pharmacokinetic and Statistical Considerations' (DIA 1991) were the logarithmic transformation of bioequivalence characteristics and the corresponding bioequivalence range. Other controversies concerning the design of bioequivalence studies are: selection of the study population (healthy subjects/ patients, smokers/nonsmokers, normal metabolisers/fast and poor metabolisers); posture (supine/ mobile); fasting or fed conditions and time between drug administration and meal intake; volume of fluid given after drug administration; the selection of the appropriate reference formulation; single vs multiple doses; and the replicated admin-
istration of test and reference formulations in the same study participant for 2 periods each. This last would allow investigation of the reproducibility of test and reference formulation, i.e. a comparison of the within-subject variability of test and reference. This has to be seen in connection with the most recent controversy in bioequivalence assessment, namely, whether average bioequivalence is a sufficient criterion to allow interchangeability of generic drugs, or whether the more stringent individual bioequivalence should become a necessary condition (Anderson & Hauck 1990). The option to choose between pharmacokinetic characteristics for rate and extent of absorption has attracted considerable attention from pharmacokineticists, particularly with regard to controlled released formulations (e.g. Steinijans 1990). During the recent DIA workshop it appeared, however, that speakers ofthe US Food and Drug Administration (FDA) still consider area under the plasma concentration-time curve (AUC) and peak concentration (C max ) as the primary characteristics of extent and rate of absorption, although the limitations of Cmax as a rate characteristic were admitted, and despite other rate characteristics such as the peaktrough fluctuation having been recommended in earlier FDA guidance (Skelly 1984). If pharmacokinetic characteristics cannot be determined, e.g. due to lack of assay sensitivity, pharmacodynamic characteristics may be used; but possible implications for the design of the study (e.g. parallel groups vs crossover) should also be considered.
248
The statistical analysis of the 2-period crossover design has been consolidated in recent years, but is still under discussion for multiperiod crossover designs including the replication of test and reference formulations. Recommendations on whether or not to adjust the significance level in the case of multiple comparisons, e.g. 3 generic test products vs 1 reference, are still lacking. The acceptance of non parametric statistical methods is not yet universal. It is the purpose of this article to discuss the abovementioned controversies, to give recommendations and to express reservations where necessary.
1. The Basic Concepts 0/ Average and Individual Bioequivalence The term bioequivalence applied to different formulations of the same drug refers to equivalence with respect to rate and extent of absorption. Two formulations in which these parameters differ by 20% or less are considered to be bioequivalent (FDA 1977). Thus, a test (T) and a reference (R) formulation will be considered bioequivalent if, for the respective pharmacokinetic characteristics, the location parameters (expected means or medians) J.lT and J.lR differ by no more than 20% of that of the reference. This statement refers to the location parameters and, therefore, the term 'average bioequivalence' is most appropriate. It was recently pointed out by Anderson and Hauck (1990) that average bioequivalence may not be sufficient to provide reasonable assurance that an individual patient could be switched from a therapeutically successful reference formulation to the new test formulation, e.g. a generic substitute. To this end a different notion of bioequivalence, referred to as individual (or within-subject) bioequivalence, is needed. The suggested decision rule Test of Individual Equivalence Ratios (TIER) requires the specification of the minimum proportion of the individuals in the population for which the individual bioavailability ratios fall within the specified bioequivalence range. Thus, TIER is based on direct consideration of the individual bioavailability ratios, as was the FDA's discontinued '75/15' rule;
Clin. Pharmacokinet. 22 (4) 1992
although TIER is a statistically valid procedure, it does not permit unequal period effects in the crossover design. Another approach which goes beyond average bioequivalence considers, in addition to the equivalence of the location parameters, the equivalence ofthe within-subject variabilities (dispersion parameters). As far as the estimation and comparison of the within-subject variability is concerned, replication of test and reference formulations in the same study participant is necessary. Apart from the 4-period crossover design with the 4 treatment sequences RRTT, RTTR, TRRT, TTRR, Schuirmann (1990) discussed a 3-period crossover design with treatment sequences RTR and TRT, and a 2period crossover design with treatment sequences RR, RT, TR and TT, because they also permit the assessment of potential subgroups in the underlying population (Ekbohm & Melander 1989).
2. Study Design For many drugs a great between-subject variability in clearance is observed. The within-subject coefficient of variation (e.g. 15%) is usually substantially smaller than that between subjects (e.g. 30%) and, therefore, crossover designs are generally recommended for bioequivalence studies. If the investigated drug and/or its metabolites have an extremely long half-life, a parallel group design may be indicated. This may also be the case in pharmacokinetic studies in patients where repeated pharmacokinetic profiles are difficult to obtain and where residual pharmacodynamic effects may be relevant. The controversy about single- and/or multipledose studies was addressed during 2 recent workshops which were organised by the International Association for Pharmaceutical Technology (APV) and the Central Laboratories of German Pharmacists (ZL) in October 1990 in Cologne (Oosterhuis & Jonkman 1991), and by the DIA in August 1991 in Bethesda, Maryland (DIA 1991 ). Single-dose studies are considered to be sufficient for comparing immediate release formulations, during the development of controlled release formulations (Be-
Controversies in Bioequivalence Studies
nedikt et a1. 1988) and to investigate the effects of food (Karim et a1. 1985a,b; Schulz et a1. 1987). With regard to food effects on controlled-release formulations, Blume et a1. (1989) proposed that bioequivalence be shown between test and reference under fasting and fed conditions, and also considered steady-state studies. They proposed a 4-period crossover study or two 2-period crossover studies, unless the lack of food effects has been proven for the reference formulation, in which case a 3-period crossover study would be sufficient with the reference being given under only I set of conditions. Such studies are definitely necessary, e.g. in the case of sustained release theophylline formulations where extreme food effects in both directions (decrease and increase of bioavailability) have been demonstrated (for a review see 10nkman 1989). It is still a matter of controversy whether they are necessary for controlled release formulations in general. In the case of controlled release formulations, single- and multiple-dose studies have been requested by regulatory authorities (e.g. Skelly 1984). Combined single- and multiple-dose studies allow the detection of nonlinear pharmacokinetics after repeated administration. Multiple-dose studies better reflect clinical reality and are often easier to perform because measurements are needed over only 1 dosage interval in steady-state. This may avoid analytical problems and those of AVC extrapolation frequently encountered in single-dose studies. In multiple-dose studies, the serum concentration-time profiles should be measured over 24h in order to account for possible day and night variations. The 24h interval between measurements is also mandatory if a controlled release formulation is compared with an immediate release formulation under steady-state conditions, and if the dosage intervals are not equal (APV 1987).
3. Pharmacokinetic Characteristics of Rate and Extent of Absorption It is generally accepted that - with respect to serum concentrations - the AVC characterises the extent of absorption. This applies to single- and
249
multiple-dose studies, as well as to immediate and controlled release formulations. The choice of an appropriate pharmacokinetic characteristic for the rate of absorption is still being discussed with great controversy (Ahr & Schaefer 1991; Blume et a1. 1989; Steinijans 1990). This is not surprising, since the time course of absorption is a fairly complex process, particularly in the case of controlled release formulations. Complete information on this process may be retained in a deconvoluted absorption profile which, however, is difficult to obtain and even more difficult to compare for bioequivalence. Another way of characterising the rate of absorption would be some integral measurement such as the mean absorption time (MAT). In both cases, an additional reference experiment with an oral solution or an intravenous infusion is needed. This separate period is usually not included in bioequivalence studies, and the mean residence time (MRT) is chosen as a substitute. However, for drugs which have a long elimination half-life, the MRT will be dominated by that factor and, therefore, is less suitable to characterise the rate of absorption. Although single-point characteristics such as Cmax and time to Cmax (tmax ) are only the third choice, they have traditionally been requested as rate characteristics by most regulatory authorities. Both t max and Cmax are strongly dependent on the discrete sampling scheme; and both are of fairly limited value in the case of controlled release formulations with their flat and sometimes multiple peaks (Boxenbaum 1984; Steinijans et a1. 1987). Therefore, the plateau time has frequently been used as a steady-state rate characteristic for controlled release formulations. The plateau time is defined as the time span of I dosage interval or dosage cycle, e.g. 24h, during which the serum concentrations deviate from the maximum concentration by less than a clinically specified difference or percentage. In the case of a 50% deviation from the maximum, the plateau time corresponds to the halfvalue duration (HVD) introduced by Meier et al. (1974). In the case of controlled-release theophylline formulations, for example, the plateau time refers to the time during which the concentration ex-
250
ceeds 75% of the maximum and, therefore, is denoted by T75%C max (Jonkman et al. 1981; Steinijans et al. 1987). A similar concept led to the 'time above the average concentration at steadystate' as a rate characteristic (Ahr & Schaefer 1991); however, we found that this characteristic has little power to discriminate between formulations. This also applies to the %AUC-fluctuation = 100· (AUC between C(t) and Cav)/AUC, which was introduced by Boxenbaum (1984) as a robust steadystate rate characteristic. The AUC-fluctuation relates the area between the measured concentrationtime curve C(t) and the horizontal line Cay, which represents the average steady-state concentration, to the total AUC during I dosage interval or dosage cycle. We also routinely calculate the %Swing = 100· (C max - Cmin)/Cmin, which relates the difference between maximum and minimum concentration to the minimum concentration. We found this characteristic to be extremely sensitive to changes and/or errors in Cmin (Steinijans et al. 1987). We therefore prefer the percentage peaktrough fluctuation (%PTF) as the steady-state rate characteristic: %PTF = 100· (C max - Cmin)/Cav , Referencing Cmax - Cmin to the extent characteristic Cay = AUC/dose interval provides largely independent rate and extent characteristics, namely %PTF and AUC (Schulz & Steinijans 1991). Similar thoughts led to the proposal to replace Cmax by Cmax/AUC in single-dose studies (Endrenyi et al. 1991). It has become apparent that the rate characteristic which is usually requested, Cmax , has substantial drawbacks, and that a variety of alternatives exist. Preferences for one or the other may depend on the pharmacokinetic profiles to be expected and even on personal experience. We consider it to be 'good biometrical practice' to specify the primary rate characteristic for confirmative bioequivalence analysis in the study protocol before beginning the bioequivalence study. We strongly object to the a posteriori selection of a rate characteristic which yields the most narrow confidence interval.
Clin. Pharmacokinet. 22 (4) 1992
4. Statistical Analysis 4.1 Additive Versus Multiplicative Model There are 3 principal strategies to be pursued: (a) to routinely use the original scale; (b) to routinely use the logarithmic transformation; and (c) to use the original or the log-scale on a case-bycase basis. Schuirmann (1991), too, recently recommended the use of the logarithmic transformation for AUC and Cmax . The argument is as follows: In the crossover design, the usual assumption is that the observation is a function of additive effects due to subject, period and formulation. Taking logarithms transforms fundamental pharmacokinetic equations of multiplicative character into additive model equations. For example: (I) AUC = clearance-I. f· dose where 0 < f ~ I denotes the fraction absorbed, is transformed into: (2) In AUC = -In clearance
+ In f + In dose
where In denotes the natural logarithm. Thus, In AUC is an additive function of the formulation effect 'In f and the subject effect '-In clearance' (Steinijans & Hauschke 1990). Similar reasons can be given for the logarithmic transformation of drug concentration data (Westlake 1988). Therefore, we recommend a logarithmic transformation for the characteristics AUC, Cmax, MRT, peak-trough fluctuation and AUC-fluctuation. In the case of t max, HVD, plateau time and the time above the average concentration, the assumption of an additive model seems appropriate; in this case the statistical analysis should be performed on the untransformed observations. 4.2 Methods of Bioequivalence Assessment The statistical analysis of 2-period crossover bioequivalence studies has been consolidated in recent years through the work ofSchuirmann (1987), Westlake (1988) and Hauschke et al. (1990). As the consumer risk of erroneously accepting bioequivalence is of primary concern for health authorities, only statistical procedures not exceeding a
Controversies in Bioequivalence Studies
251
nominal consumer risk of 5% are acceptable, and among those the one which minimises the producer risk of erroneously rejecting bioequivalence has to be selected as the decision procedure of choice. Therefore, the test procedure by Anderson and Hauck (1983) and Westlake's (1976) method using symmetrical confidence intervals cannot be recommended. The recommended decision procedure in favour of bioequivalence is based on the inclusion of the shortest 90% confidence interval for the ratio (difference) of expected medians in the respective bioequivalence range, assuming a multiplicative (additive) model. This decision is equivalent to the rejection of 2 one-sided hypotheses by means of ttests in the parametric approach (Schuirmann 1987) or by Wilcoxon tests in the corresponding nonparametric analysis (Hauschke et al. 1990). However, as weaker model assumptions are needed, the nonparametric procedure is an alternative if the assumption of a (logarithmic) normal distribution for the untransformed characteristics is doubtful. The producer risk can be controlled by appropriate sample size calculations. These should be performed before the start of the study and should be stipulated in the trial protocol. For the recommended decision procedure, tables and nomograms were provided by Phillips (1990) for the additive and by Diletti et al. (1991) for the multiplicative model. If a multiplicative model is assumed, and if ~T and ~R denote the expected medians for test and reference, respectively, the test problem can be described as follows: (3) Ho: H/~R ~ 8) or (bioinequi valence) H): 8)
Clin. Pharmacokinet. 22 (4): 247-253. 1992 0312-5963/ 92/0004-0247/$03.50/0 © Adis International Limited. All rights reserved. CPKI
Controversies in Bioequivalence Studies Volker W. Steinijans, Dieter Hauschke and Jan H.G. Jonkman Department of Biometry, Byk Gulden Pharmaceuticals, Konstanz, Germany, and Institute for Clinical Pharmacology, Pharma Bio-Research Int. B.V., Science Park, Zuidlaren, The Netherlands
The design, performance and evaluation of bioequivalence studies have received major attention from academia, the pharmaceutical industry and health authorities over the past decade. Despite the efforts of various regional and national bodies in setting up guidelines and recommendations (APV 1987; CPMP 1991; FDA 1977, 1988; Nordic Council of Medicines 1987; Skelly 1984), a universal standard is still lacking. Such a standard is becoming ever more necessary, in order to avoid costly and unethical replications of largely identical studies. Although the requests by the various health authorities may appear to differ only marginally, they may lead to different conclusions with regard to approval or rejection of bioequivalence_ Among the issues discussed during a recent workshop of the Drug Information Association (DIA) on 'Bioavailability/Bioequivalence: Pharmacokinetic and Statistical Considerations' (DIA 1991) were the logarithmic transformation of bioequivalence characteristics and the corresponding bioequivalence range. Other controversies concerning the design of bioequivalence studies are: selection of the study population (healthy subjects/ patients, smokers/nonsmokers, normal metabolisers/fast and poor metabolisers); posture (supine/ mobile); fasting or fed conditions and time between drug administration and meal intake; volume of fluid given after drug administration; the selection of the appropriate reference formulation; single vs multiple doses; and the replicated admin-
istration of test and reference formulations in the same study participant for 2 periods each. This last would allow investigation of the reproducibility of test and reference formulation, i.e. a comparison of the within-subject variability of test and reference. This has to be seen in connection with the most recent controversy in bioequivalence assessment, namely, whether average bioequivalence is a sufficient criterion to allow interchangeability of generic drugs, or whether the more stringent individual bioequivalence should become a necessary condition (Anderson & Hauck 1990). The option to choose between pharmacokinetic characteristics for rate and extent of absorption has attracted considerable attention from pharmacokineticists, particularly with regard to controlled released formulations (e.g. Steinijans 1990). During the recent DIA workshop it appeared, however, that speakers ofthe US Food and Drug Administration (FDA) still consider area under the plasma concentration-time curve (AUC) and peak concentration (C max ) as the primary characteristics of extent and rate of absorption, although the limitations of Cmax as a rate characteristic were admitted, and despite other rate characteristics such as the peaktrough fluctuation having been recommended in earlier FDA guidance (Skelly 1984). If pharmacokinetic characteristics cannot be determined, e.g. due to lack of assay sensitivity, pharmacodynamic characteristics may be used; but possible implications for the design of the study (e.g. parallel groups vs crossover) should also be considered.
248
The statistical analysis of the 2-period crossover design has been consolidated in recent years, but is still under discussion for multiperiod crossover designs including the replication of test and reference formulations. Recommendations on whether or not to adjust the significance level in the case of multiple comparisons, e.g. 3 generic test products vs 1 reference, are still lacking. The acceptance of non parametric statistical methods is not yet universal. It is the purpose of this article to discuss the abovementioned controversies, to give recommendations and to express reservations where necessary.
1. The Basic Concepts 0/ Average and Individual Bioequivalence The term bioequivalence applied to different formulations of the same drug refers to equivalence with respect to rate and extent of absorption. Two formulations in which these parameters differ by 20% or less are considered to be bioequivalent (FDA 1977). Thus, a test (T) and a reference (R) formulation will be considered bioequivalent if, for the respective pharmacokinetic characteristics, the location parameters (expected means or medians) J.lT and J.lR differ by no more than 20% of that of the reference. This statement refers to the location parameters and, therefore, the term 'average bioequivalence' is most appropriate. It was recently pointed out by Anderson and Hauck (1990) that average bioequivalence may not be sufficient to provide reasonable assurance that an individual patient could be switched from a therapeutically successful reference formulation to the new test formulation, e.g. a generic substitute. To this end a different notion of bioequivalence, referred to as individual (or within-subject) bioequivalence, is needed. The suggested decision rule Test of Individual Equivalence Ratios (TIER) requires the specification of the minimum proportion of the individuals in the population for which the individual bioavailability ratios fall within the specified bioequivalence range. Thus, TIER is based on direct consideration of the individual bioavailability ratios, as was the FDA's discontinued '75/15' rule;
Clin. Pharmacokinet. 22 (4) 1992
although TIER is a statistically valid procedure, it does not permit unequal period effects in the crossover design. Another approach which goes beyond average bioequivalence considers, in addition to the equivalence of the location parameters, the equivalence ofthe within-subject variabilities (dispersion parameters). As far as the estimation and comparison of the within-subject variability is concerned, replication of test and reference formulations in the same study participant is necessary. Apart from the 4-period crossover design with the 4 treatment sequences RRTT, RTTR, TRRT, TTRR, Schuirmann (1990) discussed a 3-period crossover design with treatment sequences RTR and TRT, and a 2period crossover design with treatment sequences RR, RT, TR and TT, because they also permit the assessment of potential subgroups in the underlying population (Ekbohm & Melander 1989).
2. Study Design For many drugs a great between-subject variability in clearance is observed. The within-subject coefficient of variation (e.g. 15%) is usually substantially smaller than that between subjects (e.g. 30%) and, therefore, crossover designs are generally recommended for bioequivalence studies. If the investigated drug and/or its metabolites have an extremely long half-life, a parallel group design may be indicated. This may also be the case in pharmacokinetic studies in patients where repeated pharmacokinetic profiles are difficult to obtain and where residual pharmacodynamic effects may be relevant. The controversy about single- and/or multipledose studies was addressed during 2 recent workshops which were organised by the International Association for Pharmaceutical Technology (APV) and the Central Laboratories of German Pharmacists (ZL) in October 1990 in Cologne (Oosterhuis & Jonkman 1991), and by the DIA in August 1991 in Bethesda, Maryland (DIA 1991 ). Single-dose studies are considered to be sufficient for comparing immediate release formulations, during the development of controlled release formulations (Be-
Controversies in Bioequivalence Studies
nedikt et a1. 1988) and to investigate the effects of food (Karim et a1. 1985a,b; Schulz et a1. 1987). With regard to food effects on controlled-release formulations, Blume et a1. (1989) proposed that bioequivalence be shown between test and reference under fasting and fed conditions, and also considered steady-state studies. They proposed a 4-period crossover study or two 2-period crossover studies, unless the lack of food effects has been proven for the reference formulation, in which case a 3-period crossover study would be sufficient with the reference being given under only I set of conditions. Such studies are definitely necessary, e.g. in the case of sustained release theophylline formulations where extreme food effects in both directions (decrease and increase of bioavailability) have been demonstrated (for a review see 10nkman 1989). It is still a matter of controversy whether they are necessary for controlled release formulations in general. In the case of controlled release formulations, single- and multiple-dose studies have been requested by regulatory authorities (e.g. Skelly 1984). Combined single- and multiple-dose studies allow the detection of nonlinear pharmacokinetics after repeated administration. Multiple-dose studies better reflect clinical reality and are often easier to perform because measurements are needed over only 1 dosage interval in steady-state. This may avoid analytical problems and those of AVC extrapolation frequently encountered in single-dose studies. In multiple-dose studies, the serum concentration-time profiles should be measured over 24h in order to account for possible day and night variations. The 24h interval between measurements is also mandatory if a controlled release formulation is compared with an immediate release formulation under steady-state conditions, and if the dosage intervals are not equal (APV 1987).
3. Pharmacokinetic Characteristics of Rate and Extent of Absorption It is generally accepted that - with respect to serum concentrations - the AVC characterises the extent of absorption. This applies to single- and
249
multiple-dose studies, as well as to immediate and controlled release formulations. The choice of an appropriate pharmacokinetic characteristic for the rate of absorption is still being discussed with great controversy (Ahr & Schaefer 1991; Blume et a1. 1989; Steinijans 1990). This is not surprising, since the time course of absorption is a fairly complex process, particularly in the case of controlled release formulations. Complete information on this process may be retained in a deconvoluted absorption profile which, however, is difficult to obtain and even more difficult to compare for bioequivalence. Another way of characterising the rate of absorption would be some integral measurement such as the mean absorption time (MAT). In both cases, an additional reference experiment with an oral solution or an intravenous infusion is needed. This separate period is usually not included in bioequivalence studies, and the mean residence time (MRT) is chosen as a substitute. However, for drugs which have a long elimination half-life, the MRT will be dominated by that factor and, therefore, is less suitable to characterise the rate of absorption. Although single-point characteristics such as Cmax and time to Cmax (tmax ) are only the third choice, they have traditionally been requested as rate characteristics by most regulatory authorities. Both t max and Cmax are strongly dependent on the discrete sampling scheme; and both are of fairly limited value in the case of controlled release formulations with their flat and sometimes multiple peaks (Boxenbaum 1984; Steinijans et a1. 1987). Therefore, the plateau time has frequently been used as a steady-state rate characteristic for controlled release formulations. The plateau time is defined as the time span of I dosage interval or dosage cycle, e.g. 24h, during which the serum concentrations deviate from the maximum concentration by less than a clinically specified difference or percentage. In the case of a 50% deviation from the maximum, the plateau time corresponds to the halfvalue duration (HVD) introduced by Meier et al. (1974). In the case of controlled-release theophylline formulations, for example, the plateau time refers to the time during which the concentration ex-
250
ceeds 75% of the maximum and, therefore, is denoted by T75%C max (Jonkman et al. 1981; Steinijans et al. 1987). A similar concept led to the 'time above the average concentration at steadystate' as a rate characteristic (Ahr & Schaefer 1991); however, we found that this characteristic has little power to discriminate between formulations. This also applies to the %AUC-fluctuation = 100· (AUC between C(t) and Cav)/AUC, which was introduced by Boxenbaum (1984) as a robust steadystate rate characteristic. The AUC-fluctuation relates the area between the measured concentrationtime curve C(t) and the horizontal line Cay, which represents the average steady-state concentration, to the total AUC during I dosage interval or dosage cycle. We also routinely calculate the %Swing = 100· (C max - Cmin)/Cmin, which relates the difference between maximum and minimum concentration to the minimum concentration. We found this characteristic to be extremely sensitive to changes and/or errors in Cmin (Steinijans et al. 1987). We therefore prefer the percentage peaktrough fluctuation (%PTF) as the steady-state rate characteristic: %PTF = 100· (C max - Cmin)/Cav , Referencing Cmax - Cmin to the extent characteristic Cay = AUC/dose interval provides largely independent rate and extent characteristics, namely %PTF and AUC (Schulz & Steinijans 1991). Similar thoughts led to the proposal to replace Cmax by Cmax/AUC in single-dose studies (Endrenyi et al. 1991). It has become apparent that the rate characteristic which is usually requested, Cmax , has substantial drawbacks, and that a variety of alternatives exist. Preferences for one or the other may depend on the pharmacokinetic profiles to be expected and even on personal experience. We consider it to be 'good biometrical practice' to specify the primary rate characteristic for confirmative bioequivalence analysis in the study protocol before beginning the bioequivalence study. We strongly object to the a posteriori selection of a rate characteristic which yields the most narrow confidence interval.
Clin. Pharmacokinet. 22 (4) 1992
4. Statistical Analysis 4.1 Additive Versus Multiplicative Model There are 3 principal strategies to be pursued: (a) to routinely use the original scale; (b) to routinely use the logarithmic transformation; and (c) to use the original or the log-scale on a case-bycase basis. Schuirmann (1991), too, recently recommended the use of the logarithmic transformation for AUC and Cmax . The argument is as follows: In the crossover design, the usual assumption is that the observation is a function of additive effects due to subject, period and formulation. Taking logarithms transforms fundamental pharmacokinetic equations of multiplicative character into additive model equations. For example: (I) AUC = clearance-I. f· dose where 0 < f ~ I denotes the fraction absorbed, is transformed into: (2) In AUC = -In clearance
+ In f + In dose
where In denotes the natural logarithm. Thus, In AUC is an additive function of the formulation effect 'In f and the subject effect '-In clearance' (Steinijans & Hauschke 1990). Similar reasons can be given for the logarithmic transformation of drug concentration data (Westlake 1988). Therefore, we recommend a logarithmic transformation for the characteristics AUC, Cmax, MRT, peak-trough fluctuation and AUC-fluctuation. In the case of t max, HVD, plateau time and the time above the average concentration, the assumption of an additive model seems appropriate; in this case the statistical analysis should be performed on the untransformed observations. 4.2 Methods of Bioequivalence Assessment The statistical analysis of 2-period crossover bioequivalence studies has been consolidated in recent years through the work ofSchuirmann (1987), Westlake (1988) and Hauschke et al. (1990). As the consumer risk of erroneously accepting bioequivalence is of primary concern for health authorities, only statistical procedures not exceeding a
Controversies in Bioequivalence Studies
251
nominal consumer risk of 5% are acceptable, and among those the one which minimises the producer risk of erroneously rejecting bioequivalence has to be selected as the decision procedure of choice. Therefore, the test procedure by Anderson and Hauck (1983) and Westlake's (1976) method using symmetrical confidence intervals cannot be recommended. The recommended decision procedure in favour of bioequivalence is based on the inclusion of the shortest 90% confidence interval for the ratio (difference) of expected medians in the respective bioequivalence range, assuming a multiplicative (additive) model. This decision is equivalent to the rejection of 2 one-sided hypotheses by means of ttests in the parametric approach (Schuirmann 1987) or by Wilcoxon tests in the corresponding nonparametric analysis (Hauschke et al. 1990). However, as weaker model assumptions are needed, the nonparametric procedure is an alternative if the assumption of a (logarithmic) normal distribution for the untransformed characteristics is doubtful. The producer risk can be controlled by appropriate sample size calculations. These should be performed before the start of the study and should be stipulated in the trial protocol. For the recommended decision procedure, tables and nomograms were provided by Phillips (1990) for the additive and by Diletti et al. (1991) for the multiplicative model. If a multiplicative model is assumed, and if ~T and ~R denote the expected medians for test and reference, respectively, the test problem can be described as follows: (3) Ho: H/~R ~ 8) or (bioinequi valence) H): 8)
