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Introduction

I have tried to show the time course of the bioequivalence assessment. Regrettably, its development proved to be a convoluted process. Some approaches were proposed, subsequently abandoned, only to reemerge decades later in a modified form. Conversely, others arose from intuitive gut feelings, devoid of any scientific foundation, yet they persist in their utilization.

If this article is perceived as overly focused on statistics, I apologize. This is due to my professional background, which has led me to be less skilled at crafting engaging narratives.
I have to confess that »Short« in the title is a euphemism…

‘Bioavailability’ (a portmanteau of ‘biologic availability’) in its current meaning was coined in 1971¹ and ‘Bioequivalence’ saw the light of day in 1975.²

The MeSH term ‘Biological Availability’ was introduced in 1979.

“The extent to which the active ingredient of a drug dosage form becomes available at the site of drug action or in a biological medium believed to reflect accessibility to a site of action.

The site of action (i.e., a receptor) is practically always inaccessible. There is no space for believes in science.

The main assumption in Bioequivalence (BE) was – and still is – that ‘similar’ concentrations in the systemic circulation of healthy volunteers will lead to similar concentrations at the target site (i.e., a receptor) and thus, to similar effects in patients.

The best definition of BE is given by the International Council for Harmonisation of Technical Requirements for Pharmaceuticals for Human Use (ICH).³

“Two drug products containing the same drug substance(s) are considered bioequivalent if their relative bioavailability (BA) (rate and extent of drug absorption) after administration in the same molar dose lies within acceptable predefined limits. These limits are set to ensure comparable in vivo performance, i.e., similarity in terms of safety and efficacy.
ICH (2020)³

Throughout the article we will use data of a study in a two-treatment two-sequence two-period (2×2×2) crossover design as an example. \[\small{\begin{array}{cccc} \textsf{Table I}\phantom{0}\\ \text{subject} & \text{sequence} & \text{T} & \text{R}\\\hline \phantom{1}1 & \text{RT} & 71 & 81\\ \phantom{1}2 & \text{TR} & 61 & 65\\ \phantom{1}3 & \text{RT} & 80 & 94\\ \phantom{1}4 & \text{TR} & 66 & 74\\ \phantom{1}5 & \text{TR} & 94 & 54\\ \phantom{1}6 & \text{RT} & 97 & 63\\ \phantom{1}7 & \text{RT} & 70 & 85\\ \phantom{1}8 & \text{TR} & 76 & 90\\ \phantom{1}9 & \text{TR} & 54 & 53\\ 10 & \text{RT} & 99 & 56\\ 11 & \text{RT} & 83 & 90\\ 12 & \text{TR} & 51 & 68\\\hline \end{array}}\]

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The 1970s

Problems were reported with formulations of Narrow Therapeutic Index Drugs (NTIDs) like phenytoin,⁴ ⁵ ⁶ ⁷ digoxin,¹ ⁸ ⁹ ¹⁰ ¹¹ ¹² warfarin,¹³ theophylline,¹⁴ primidone.¹⁵ Some show nonlinear pharmacokinetics (phenytoin) or are auto-inducers (warfarin).

Poor content uniformity¹
Excipient changed from CaS0₄ to lactose^{5 6}
API altered (e.g., particle size,^{7 12} amorphous to crystalline¹⁴)
Variable disintegration time
Dissolution testing not mandatory
No in vivo studies were performed comparing the new to the approved formulation
Breakthrough-seizures⁴ and intoxications^{5 6} (phenytoin) and variable or poor effect (digoxin, theophylline)

Generic drugs in the current sense did not yet exist at that time; only the content had to meet the USP requirements.

“Although in 1969 Professor John Wagner demonstrated to the Bureau of Medicine, methods for comparing areas under the serum versus time curve (AUC) to estimate bioequivalence, his approach was ignored inasmuch as the FDA hierarchy did not believe a problem existed, and therefore such studies would not be neccessary. For their part the Offices of Pharmaceutical Research and Compliance in the Bureau of Medicine and the Commissioner’s Office believed that the “Bioavailability Problem” as some called it was a “Content Uniformity Problem”.¹⁶ In 1971 for example, when notified of a “Bioavailability Problem” with a generic digoxin product, FDA investigated and ascertained that one manufacturer first added all the excipients into a 55-gal drum, then added digoxin, closed the lid, and mixed it by rolling the drum across the floor a few times. The content uniformity of those tablets varied from 10% to 156%.
Jerome P. Skelly (2010)¹⁷

Following a ‘Conference on Bioavailability of Drugs’ held at the National Academy of Sciences of the United States in 1971, a guideline was published the following year.¹⁸

“[…] the mean of AUC of the generic had to be within 20% of the mean AUC of the approved product. At first this was determined by using serum versus time plots on specially weighted paper, cutting the plot out and then weighing each separately.
Jerome P. Skelly (2010)¹⁷

Methods and procedures for in vivo testing to determine bioavailability (BA) for new drugs were proposed by the FDA on June 20, 1975. Several terms were defined:¹⁹

Bioavailability
The rate and extent to which the therapeutic moiety is absorbed and becomes available to the site of drug action, normally estimated by its concentrations in body fluids, rate of excretion, or acute pharmacologic effect.
Drug Product
A finished dosage form (e.g., tablet, capsule, solution) that contains the active drug ingredient often, but not necessarily, in association with inactive ingredients.
Pharmaceutical Equivalents
Drug products that contain the same quantities of the identical active drug (but not necessarily containing the same inactive) ingredient (i.e., the same salt or ester of the same therapeutic moiety) in an identical dosage form and that meet the compendial or other applicable standard of identity, strength, quality, and purity, including potency and, where applicable, content uniformity, disintegration times, and/or dissolution rates.
Pharmaceutical Alternatives
Drug products that contain the identical therapeutic moiety (or its precursor), but not necessarily in the same amount or the same dosage form, or as the same salt or ester. Each such drug product meets its own compendial or other applicable standard of strength, quality, and purity, including potency and, where applicable, content uniformity, disintegration time, and dissolution rate.
Bioequivalent Drug Products
Pharmaceutical equivalents or pharmaceutical alternatives which are not significantly different with respect to rate and extent of absorption when administered at the same molar dose under similar experimental conditions (single dose or multiple dose). Some pharmaceutical equivalents may be equivalent in the extent but not the rate of their absorption, and yet may be considered bioequivalent because the differences in rates of absorption may be considered clinically insignificant for the particular drug products studied.
Bioequivalence Requirement
A requirement, imposed by the FDA for in vitro and/or in vivo testing of specific drug products, which will be required of all manufacturers as a condition of marketing.

The “site of drug action” was questioned but kept in the regulation of 1977 and is used ever since by the FDA.²⁰

“[A] comment also recommended that the phrase “becomes available to the site of drug action” be deleted since it is overly optimistic to presume that bioavailability data consisting of estimates of parent drug […] concentration in body fluids […] provides, as a general rule, an estimate of the availability of the therapeutic moiety at the site of drug action.
The Commissioner agrees that bioavailability data alone do not estimate the availability of the therapeutic moiety at the site of drug action. It is scientifically valid to assume, however, that if an active drug ingredient or therapeutic moiety reaches a reasonable extent of systemic circulation at a reasonable rate, the therapeutic moiety will also become available at the site of drug action […]. For this reason, the Commissioner concludes that reference to availability at site of drug action should not be deleted. He also believes that omission of such a reference would incorrectly focus the definition of bioavailability exclusively on absorption of the active drug ingredient or therapeutic moiety from the drug product. Even where such absorption is total, the product may not be bioavailable because an insufficient amount of the active drug ingredient or therapeutic moiety reaches the systemic circulation. In certain instances, e.g., high first-pass metabolism in the liver or rapid renal clearance, the active drug ingredient or therapeutic moiety must be absorbed at a rate sufficient to overcome the metabolic or elimination mechanism and reach the systemic circulation so that the therapeutic moiety will become available at the site of drug action in sufficient amounts to elicit the intended therapeutic effect.
Sherwin Gardener (1976)²⁰

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80/20 Rule

The FDA’s 80/20 Rule or ‘Power Approach’ (at least 80% power to detect a 20% difference) of 1972 consisted of testing the hypothesis of no difference at the $\small{\alpha=}$ $\small{0.05}$ level of significance.¹⁷ ²¹ \[H_0:\;\mu_\text{T}-\mu_\text{R}=0\;vs\;H_1:\;\mu_\text{T}-\mu_\text{R}\neq 0,\tag{1}\] where $\small{H_0}$ is the null hypothesis of equivalence and $\small{H_1}$ the alternative hypothesis of inequivalence. $\small{\mu_\text{T}}$ and $\small{\mu_\text{R}}$ are the (true) means of $\small{\text{T}}$ and $\small{\text{R}}$, respectively. In order to pass the test, the estimated (post hoc, a posteriori, retrospective) power had to be at least 80%. The power depends on the true value of $\small{\sigma}$, which is unknown. There exists a value of $\small{\sigma_{\,0.80}}$ such that if $\small{\sigma\leq\sigma_{\,0.80}}$, the power of the test of no difference $\small{H_0}$ is greater or equal to 0.80. Since $\small{\sigma}$ is unknown, it has to be approximated by the sample standard deviation $\small{s}$. The Power Approach in a simple 2×2×2 crossover design then consists of rejecting $\small{H_0}$ and concluding that ${\small{\mu_\text{T}}}$ and ${\small{\mu_\text{R}}}$ are equivalent if \[-t_{1-\alpha/2,\nu}\leq\frac{\bar{x}_\text{T}-\bar{x}_\text{R}}{s\sqrt{\tfrac{1}{2}\left(\tfrac{1}{n_1}+\tfrac{1}{n_2}\right)}}\leq t_{1-\alpha/2,\nu}\:\textsf{and}\:s\leq\sigma_{0.80}\textsf{,}\tag{2}\] where $\small{n_1,\,n_2}$ are the number of subjects in sequences 1 and 2, the degrees of freedom $\small{\nu=n_1+n_2-2}$, and $\small{\bar{x}_\text{T},\bar{x}_\text{R}}$ are the means of $\small{\text{T}}$ and $\small{\text{R}}$, respectively.
Note that this procedure is based on estimated power $\small{\widehat{\pi}}$, since the true power is a function of the unknown $\small{\sigma}$. It was the only approach based on post hoc power and was never implemented in any other jurisdiction.

For the example we estimate a power of only 46.4% to detect a 20% difference and the study would fail.

The biostatistical community published alternative proposals:²² ²³ ²⁴ ²⁵

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95% CI

The analysis was performed on untransformed (raw) data (i.e., by an additive model assuming normal distributions) and BE was concluded if the 95% confidence interval (CI) of the point estimate (PE) lied entirely within 80 – 120%.^{22 25}

If data are analyzed by an additive model the result are differences.
It is a fundamental error to naïvely transform results to percentages (i.e., dividing by the reference’s mean). It would require Fieller’s CI for the ratio of normal distributed data.²⁶ ²⁷ However, Locke’s paper²⁷ was ignored back in the day.

We get for our example in R:

n             <- 12L
example       <- data.frame(subject   = rep(1L:n, each = 2),
                            treatment = c("R", "T", "T", "R", "R", "T", "T", "R",
                                          "T", "R", "R", "T", "R", "T", "T", "R",
                                          "T", "R", "R", "T", "R", "T", "T", "R"),
                            period    = rep(1L:2L, n),
                            Y         = c(81, 71, 61, 65, 94, 80, 66, 74,
                                          94, 54, 63, 97, 85, 70, 76, 90,
                                          54, 53, 56, 99, 90, 83, 51, 68))
facs          <- c("subject", "period", "treatment")
example[facs] <- lapply(example[facs], factor) # factorize the data
# additive model (untransformed data, differences); sequence not in the model!
muddle        <- lm(Y ~ subject + period + treatment, data = example)
CI            <- as.numeric(confint(muddle, level = 0.95)["treatmentT", ])
PE            <- coef(muddle)[["treatmentT"]]
# percentages (flawed!)
X.T           <- mean(example$Y[example$treatment == "T"])
X.R           <- mean(example$Y[example$treatment == "R"])
PE.pct        <- 100 * X.T / X.R
CI.pct        <- 100 * (CI + X.R) / X.R
# Fieller’s CI (ratio of normal distributed data)
s2.TT         <- var(example$Y[example$treatment == "T"])
s2.RR         <- var(example$Y[example$treatment == "R"])
s2.TR         <- cov(example$Y[example$treatment == "T"],
                     example$Y[example$treatment == "R"])
pe            <- X.T / X.R # same like in the additive model
t             <- qt(p = 0.025, df = n - 1, lower.tail = FALSE)
G             <- t^2 * s2.RR / (n * X.R^2)
K             <- pe^2 + s2.TT / s2.RR * (1 - G) +
                 s2.TR / s2.RR * (G * s2.TR / s2.RR - 2 * pe)
ci            <- setNames(c(100 * ((pe - G * s2.TR / s2.RR) + c(-1, 1) * t / X.R *
                                    sqrt(s2.RR / n * K)) / (1 - G)),
                          c("lower", "upper"))
result        <- data.frame(method = c("differences", "percentages", "Fieller"),
                            PE     = c(sprintf("%+.3f", PE),
                                       sprintf("%6.2f%%", PE.pct),
                                       sprintf("%6.2f%%", 100 * pe)),
                            lower  = c(sprintf("%+.3f", CI[1]),
                                       sprintf("%.2f%%", CI.pct[1]),
                                       sprintf("%.2f%%", ci[["lower"]])),
                            upper  = c(sprintf("%+.3f", CI[2]),
                                       sprintf("%6.2f%%", CI.pct[2]),
                                       sprintf("%.2f%%", ci[["upper"]])),
                            BE     = c("? ", rep("fail", 2)))
if (CI.pct[1] >= 80 & CI.pct[2] <= 120) result$BE[2] <- "pass"
if (ci[["lower"]] >= 80 & ci[["upper"]] <= 120) result$BE[3] <- "pass"
names(result)[3:4] <- c("lower CL", "upper CL")
print(result, row.names = FALSE)

#       method      PE lower CL upper CL   BE
#  differences  +2.417  -12.777  +17.611   ? 
#  percentages 103.32%   82.44%  124.21% fail
#      Fieller 103.32%   84.84%  125.65% fail

With the naïve transformation we get a 95% CI of 82.44 – 124.21%, and the study would fail because the upper confidence limit (CL) is > 120%. Nevertheless, with the correct method the study would fail as well.

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Westlake’s CI

Westlake²³ mused that the shortest CI – which is symmetrical about the PE – would be too difficult to comprehend by non-statisticians. He suggested to split the t-values in such a way that the probability of the two tails sums to $\small{\alpha}$ and the respective CI is symmetrical around 0 (or 100%). In the example we obtain ±21.80%, and the study would fail as well because the confidence limits are > ±20%. As above, calculating a percentage is flawed.

However, such a result is misleading. The information about the location of the difference is lost; one cannot know any more whether the average BA of $\small{\text{T}}$ is lower or higher than the one of $\small{\text{R}}$. Therefore, the method was criticized²⁴ and never implemented in practice. It took me years to convince Certara to remove Westlake’s CI from the results in Phoenix WinNonlin. In 2016, I was successful with version 6.4… Since then the differences are given in the additive model.

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The Roaring 1980s

The ‘Approved Drug Products with Therapeutic Equivalence Evaluations’ was published and is annually updated²⁸ with monthly supplements.²⁹ The nickname “Orange Book” relates to the color commonly associated with Halloween, which is the date of the publication’s finalization – October 31, 1980 – and to the book’s orange-colored cover. It gives information about the originator’s approval (with a ‘New Drug Application’ – NDA), as well as which originator’s product and strength (called Reference Listed Drug – RLD) has to be used in studies of generics in an ‘Abbreviated New Drug Application’ – ANDA. Generic prescription drugs are coded as follows:

Drug products that FDA considers to be therapeutically equivalent to other pharmaceutically equivalent products, i.e., drug products for which:
1. there are no known or suspected BE problems. These are designated AA, AN, AO, AP, or AT, depending on the dosage form; or
2. actual or potential BE problems problems have been resolved with adequate in vivo and/or in vitro evidence supporting BE problems. These are designated AB.
Drug products that FDA at this time, considers not to be therapeutically equivalent to other pharmaceutically equivalent products, i.e., drug products for which actual or potential BE problems have not been resolved by adequate evidence of BE. Often the problem is with specific dosage forms rather than with the active ingredients. These are designated BC, BD, BE, BN, BP, BR, BS, BT, BX, or B*.

See also information about the ‘Electronic Orange Book’ below.

The generic boom started 1984 in the U.S. with the ‘Drug Price Competition and Patent Term Restoration Act’ (informally known as ‘Hatch-Waxman Act’).³⁰

The approval process was different for innovator (originator) and generic companies.

Innovators:

Preclinical data
Documentation of pharmaceutical quality
In clinical phase I documentation of pharmacokinetics (PK) in healthy subjects, dose finding, safety / tolerability, food effect
In phase II efficacy & safety in a small groups of patients
In phase III demonstration of efficacy & safety versus placebo in well-powered studies
- Efficacy: Non-Inferiority/Superiority

Generic companies:

Documentation of pharmaceutical quality
Not required:
- Any in vivo study
- Sometimes comparison of disintegration, rarely comparison of dissolution was performed

Regulatory concerns about generic substitution arose, leading to extensive discussions which approach could be suitable to compare formulations.

Pharmaceutical equivalence
Bioequivalence (BE)
Therapeutic equivalence

There was an early agreement that pharmaceutical equivalence (content, in vitro) is too permissive and therapeutic equivalence (like in phase III) would require extremely large studies in patients.³¹ Hence, comparing BA in healthy volunteers seemed to be a reasonable compromise.³²

“What is the justification for studying bioequivalence in healthy volunteers?
“Variability is the enemy of therapeutics” and is also the enemy of bioequivalence. We are trying to determine if two dosage forms of the same drug behave similarly. Therefore we want to keep any other variability not due to the dosage forms at a minimum. We choose the least variable “test tube”, that is, a healthy volunteer.
Disease states can definitely change bioavailability, but we are testing for bioequivalence, not bioavailability.
Leslie Z. Benet (2013)³²

Whereas in PK by bioavailability exclusively the Area under Curve extrapolated to infinite time $\small{(AUC_{0-\infty}})$ is meant, the FDA introduced in 1975 two new terms, namely

Therefore, PK metrics, whereas PK parameters refer to modeling.

the ‘rate of bioavailability’ (peak exposure) – measured by the maximum concentration $\small{(C_\text{max}})$ and
the ‘extent of bioavailability’ (total exposure) – measured by the $\small{AUC}$.

The former is understood as a surrogate for the absorption rate $\small{k\,_\text{a}}$ in a PK model. I prefer – like the ICH³ and the FDA since 2003³³ – rate and extent of absorption, in order not to contaminate the original meaning of BA in PK. Whereas the FDA and China’s CDE require for single dose studies $\small{AUC_{0-\text{t}}}$ and $\small{AUC_{0-\infty}}$, in all other jurisdictions only $\small{AUC_{0-\text{t}}}$ is required.

Let us consider the basic equation of pharmacokinetics \[\eqalign{ \frac{f\cdot D}{CL}&=\frac{f\cdot D}{V\cdot k_\text{el}}=\\ AUC_{0-\infty}&=\int_{0}^{\infty}C(t)\,dt\textsf{,}\tag{3}}\]

where $\small{f}$ is the fraction absorbed (we are interested in the comparison of formulations), $\small{D}$ is the dose, $\small{CL}$ is the clearance, $\small{V}$ is the apparent volume of distribution, $\small{k\,_\text{el}}$ is the elimination rate constant, and $\small{C(t)}$ is the plasma concentration with time. We see immediately that for identical³⁴ doses and invariate³⁵ $\small{CL}$, $\small{V}$, $\small{k\,_\text{el}}$ (which are drug-specific), comparing the $\small{AUC}\text{s}$ allows to compare the fractions absorbed.
Note that the top row of $\small{(3)}$ is for a one-compartment model. Nevertheless, the bottom row is universally valid, i.e., for any number of compartments and absorption ($\small{k\,_\text{a}}$, eventual lag-time) is irrelevant.

“Pharmacokinetics: one of the magic arts of divination whereby needles are stuck into dummies in an attempt to predict profits.
Stephen Senn (2004)

It must be mentioned that $\small{C_\text{max}}$ is not sensitive to even substantial changes in the rate of absorption $\small{k\,_\text{a}}$, since it is a composite metric.³⁶ In a one compartment model it depends on $\small{k\,_\text{a}}$, $\small{f}$ and both the elimination rate constant $\small{k\,_\text{el}}$ and $\small{V}$ (or $\small{CL}$ if you belong to the other church). Whereas $\small{k\,_\text{a}}$ and $\small{f}$ are properties of the formulation – we are interested in – the others are properties of the drug.³⁷ \[\eqalign{ t_\textrm{max}&=\frac{\log_{e}(k\,_\text{a}/k\,_\text{el})}{k\,_\text{a}-k\,_\text{el}}\\ C_\textrm{max}&=\frac{f\cdot D\cdot k\,_\text{a}}{V\cdot (k\,_\text{a}-k\,_\text{el})}\large(\small\exp(-k\,_\text{el}\cdot t_\textrm{max})-\exp(-k\,_\text{a}\cdot t_\textrm{max})\large)\tag{4}}\] Therefore, when using it as a surrogate for the absorption rate one must keep in mind that formulations with different fractions absorbed and absorption rate constants will show a T/R-ratio of $\small{C_\text{max}}$ which differs from the one of $\small{AUC}$ (which is independent from $\small{k\,_\text{a}}$ and thus, unbiased with regard to $\small{f}$).

$Fig. 1 Formulations with different \small{f} and \small{k\,_\text{a}}.$

Fig. 1 Formulations with different $\small{f}$ and $\small{k\,_\text{a}}$.

An assessment of $\small{t_\text{max}}$ would not necessarily help; in this example it is 2.71 h for the formulation with $\small{k\,_\text{a}=}$ $\small{0.74}$ / h and 2.78 h for the one with $\small{k\,_\text{a}=}$ $\small{0.71}$ / h. A difference of only four minutes cannot be detected with common sampling schedules.

It took ten years before the alternative metric $\small{C_\text{max}/AUC}$ (based on theoretical considerations and simulations) was proposed.³⁸ ³⁹ ⁴⁰ Apart from being less biased than $\small{C_\text{max}}$, it is also substantially less variable. Regrettably, it was never implemented in any guideline.

In the early 1980s originators failed in trying to falsify the concept (i.e., comparing BE in healthy volunteers to large therapeutic equivalence (TE) studies in patients): If BE passed, TE passed as well and vice versa. If they would have succeeded (BE passed while TE failed), generic companies would have to demonstrate TE in order to get products approved. Such studies would have to be much larger than the originators’ phase III studies, making them economically infeasible.³¹ Essentially, that would have meant an early end of the young generic industry.

However, comparative BA is also used by originators in scale-up of formulations used in phase III to the to-be-marketed formulation, supporting post-approval changes, in line extensions of approved products, and for testing of drug-drug interactions or food effects. Hence, a substantial part of BE trials are performed by originators. If they had been successful to refute the concept, they would have shot into their own foot.

In the mid 1980s a consensus was reached, i.e., that generic approval should only be acceptable after suitable in vivo equivalence. It must be mentioned that BE relies on current Good Manufacturing Practices (cGMP). If drugs are not manufactured according to cGMP, the entire concept would collapse.

It was an open issue whether BE should be interpreted as a surrogate of clinical efficacy / safety or a measure of pharmaceutical quality. Whereas in the 1980s the former was prevalent, since the 1990s the latter is mainstream.
A somewhat naïve interpretation of the PK metrics is that $\small{AUC}$ directly translates to efficacy and $\small{C_\text{max}}$ to safety. Especially the latter is not correct because any difference in $\small{C_\text{max}}$ leads to a relatively smaller difference in the maximum effect $\small{E_\text{max}}$.

There was no consensus about the definition of ‘similarity’ and the statistical methodology to compare plasma profiles. Two early methods are outlined in the following.

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75/75 Rule

This was an approach employed by the FDA. Two drugs were considered bioequivalent if at least 75% of subjects show $\small{\text{T}/\text{R}\textsf{-}}$ratios within 75 – 125%.¹⁷ ⁴³ ⁴⁴ It is not a statistic and, thus, was immediately criticized because variable formulations or studies with some extreme values may pass the criterion by pure chance.⁴⁵

We get for our example in R:

example       <- data.frame(subject   = rep(1:12, each = 2),
                            treatment = c("R", "T", "T", "R", "R", "T", "T", "R",
                                          "T", "R", "R", "T", "R", "T", "T", "R",
                                          "T", "R", "R", "T", "R", "T", "T", "R"),
                            Y         = c(81, 71, 61, 65, 94, 80, 66, 74,
                                          94, 54, 63, 97, 85, 70, 76, 90,
                                          54, 53, 56, 99, 90, 83, 51, 68))
rule.75.75    <- reshape(example, idvar = "subject", timevar = "treatment",
                         direction = "wide")
rule.75.75    <- rule.75.75[c("subject", "Y.T", "Y.R")]
names(rule.75.75)[2:3] <- c("T", "R")
rule.75.75$T.R <- 100 * (rule.75.75$T / rule.75.75$R)
for (i in 1:nrow(rule.75.75)) {
  if (rule.75.75$T.R[i] >= 75 & rule.75.75$T.R[i] <= 125) {
    rule.75.75$BE[i]     <- TRUE
    rule.75.75$within[i] <- "yes"
  } else {
    rule.75.75$BE[i]     <- FALSE
    rule.75.75$within[i] <- "no"
  }
}
names(rule.75.75)[c(4, 6)] <- c("T/R (%)", "±25%")
if (sum(rule.75.75$BE) / nrow(rule.75.75) >= 0.75) {
  BE <- "Passed BE by the"
} else {
  BE <- "Failed BE by the"
}
print(rule.75.75[, c(1:4, 6)], row.names = FALSE); cat(BE, "75/75 Rule.\n")

#  subject  T  R   T/R (%) ±25%
#        1 71 81  87.65432  yes
#        2 61 65  93.84615  yes
#        3 80 94  85.10638  yes
#        4 66 74  89.18919  yes
#        5 94 54 174.07407   no
#        6 97 63 153.96825   no
#        7 70 85  82.35294  yes
#        8 76 90  84.44444  yes
#        9 54 53 101.88679  yes
#       10 99 56 176.78571   no
#       11 83 90  92.22222  yes
#       12 51 68  75.00000  yes
# Passed BE by the 75/75 Rule.

Nine of the twelve subjects (75%) have a $\small{\text{T}/\text{R}\textsf{-}}$ratio within 75 – 125% and the study would pass, despite the three subjects with high $\small{\text{T}/\text{R}\textsf{-}}$ratios.

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t-test

Another suggestion was testing for a statistically significant difference at level $\small{\alpha=0.05}$ with a t-test. The null hypothesis was that formulations are equal, i.e., $\small{\mu_\text{T}-\mu_\text{R}=0}$.

Let’s assess our example in R again:

example        <- data.frame(subject   = rep(1:12, each = 2),
                             treatment = c("R", "T", "T", "R", "R", "T", "T", "R",
                                           "T", "R", "R", "T", "R", "T", "T", "R",
                                           "T", "R", "R", "T", "R", "T", "T", "R"),
                             Y         = c(81, 71, 61, 65, 94, 80, 66, 74,
                                           94, 54, 63, 97, 85, 70, 76, 90,
                                           54, 53, 56, 99, 90, 83, 51, 68))
tt             <- reshape(example, idvar = "subject", timevar = "treatment",
                          direction = "wide")
tt             <- tt[c("subject", "Y.T", "Y.R")] # change for clarity
tt$T.R         <- tt[, 2] - tt[, 3]              # difference
names(tt)[2:4] <- c("T", "R", "T–R")             # cosmetics
tt[, 4]        <- sprintf("%+0.f", tt[, 4])
p              <- t.test(x = tt$T, y = tt$R, paired = TRUE)$p.value
if (p >= 0.05) {BE <- "Passed BE" } else {BE<- "Failed BE" }
print(tt, row.names = FALSE); cat(sprintf("%s by a paired t-test (p = %.4f).\n", BE, p))

#  subject  T  R T–R
#        1 71 81 -10
#        2 61 65  -4
#        3 80 94 -14
#        4 66 74  -8
#        5 94 54 +40
#        6 97 63 +34
#        7 70 85 -15
#        8 76 90 -14
#        9 54 53  +1
#       10 99 56 +43
#       11 83 90  -7
#       12 51 68 -17
# Passed BE by a paired t-test (p = 0.7193).

We calculate a $\small{p}$-value of 0.7193, which is statistically not significant and the study would pass again.

However, we face a similar problem like with the 75/75 Rule. If the differences show high variability, the study would pass. On the other hand, if there is low variability in the differences, the study would fail. This is counterintuitive and actually the opposite of what regulators want.

Interlude 1

One of my early sins⁴⁶ – it was not the last…
After phenytoin intoxications in Austria⁴⁷ we compared three generics (containing the free acid like the originator, the sodium- or calcium-salt) to the reference in a crossover study. All formulations have been approved and were marketed in Austria. Although at that time I already calculated a 95% CI, the reviewers of our manuscript insisted in testing for a significant difference »because it is state of the art«.

Fig. 2 Phenytoin 3 × 100 mg equivalent, single dose fasting.

The $\small{AUC}$s of two generics were statistically significant different from the reference ($\small{\text{T}_1}$ containing the free acid like the originator and $\small{\text{T}_3}$ containing the Ca-salt). $\small{\text{T}_2}$ containing the Na-salt was statistically not significant different and, thus, considered equivalent – despite its high $\small{\text{T}/\text{R}\textsf{-}}$ratio (Table II). \[\small{ \begin{array}{ccccc} \textsf{Table II}\phantom{00000}\\ \text{formulation} & \text{T}/\text{R (%)} & p & & \text{BE}\\\hline \text{T}_1 & 146.65 & 0.0195\phantom{6} & \text{*} & \text{fail}\\ \text{T}_2 & 133.67 & 0.151\phantom{96} & \text{n.s.} & \text{pass}\\ \text{T}_3 & \phantom{1}27.97 & 0.00596 & \text{**} & \text{fail}\\\hline \end{array}}\] If we evaluate the study according to current standards (i.e., by the 90% CI inclusion approach based on $\small{\log_{e}\textsf{-}}$transformed data and acceptance limits of 80.00 – 125.00%), all generics would fail. $\small{\text{T}_3}$ would even be bioinequivalent because its upper CL is way below 80% (Table III).
\[\small{\begin{array}{ccccc} \textsf{Table III}\phantom{0000}\\ \text{formulation} & \text{PE (%)} & \text{CL}_\text{lower}\text{(%)} & \text{CL}_\text{upper}\text{ (%)} & \text{BE}\\\hline \text{T}_1 & 151.12 & 118.75 & 192.32 & \text{fail (inconclusive)}\\ \text{T}_2 & 139.39 & \phantom{1}95.91 & 202.60 & \text{fail (inconclusive)}\\ \text{T}_3 & \phantom{1}21.67 & \phantom{1}10.25 & \phantom{2}45.81 & \text{fail (inequivalent)}\\\hline \end{array}}\] Given the nonlinear PK of phenytoin,⁴⁸ ⁴⁹ switching a patient from the originator to the generics with high $\small{\text{T}/\text{R}\textsf{-}}$ratios would be problematic – potentially leading to toxicity after multiple doses. Even worse would be switching from the generic $\small{\text{T}_3}$ with its low $\small{\text{T}/\text{R}\textsf{-}}$ratio to any of the other formulations.

top of section ↩︎ previous section ↩︎

ANOVA and beyond

An Analysis of Variance (ANOVA) instead of a t-test allows to take period-effects into account.⁵⁰ ⁵¹ ⁵² This decade was also the heyday of Bayesian methods.⁵³ ⁵⁴ ⁵⁵ ⁵⁶ Nomograms for sample size estimation were also Bayesian⁵⁷ but happily misused by frequentists. New parametric ⁵⁸ ⁵⁹ as well as nonparametric methods entered the stage.⁵⁸ ⁶⁰ PK metrics to compare controlled release formulations in steady state were proposed.⁶¹ ⁶² ⁶³ The first software to evaluate 2×2×2 crossover studies was released in the public domain.⁶⁴

The acceptance range in bioequivalence is based on a ‘clinically relevant difference’ $\small{\Delta}$, i.e., for data following a lognormal distribution \[\left\{\theta_1,\theta_2\right\}=\left\{100\,(1-\Delta),100\,(1-\Delta)^{-1}\right\}\tag{5}\] It must be mentioned that the commonly applied $\small{\Delta=20\%}$⁶⁵ leading to $\small{\{80.00\%,}$ $\small{125.00\%\}}$ is arbitrary (as is any other).

An important leap forward was the Two One-Sided Tests Procedure (TOST)²¹ – although it was never implemented in its original form $\small{(6)}$ in regulatory practice. Instead, the confidence interval inclusion approach $\small{(7)}$ made it to the guidelines. Although these approaches are operationally identical (i.e., their outcomes [pass | fail] are the same), these are statistically different methods:

The TOST Procedure gives two $\small{p}$-values, namely $\small{p(\theta_0\geq\theta_1)}$ and $\small{p(\theta_0\leq\theta_2)}$.

\[\begin{matrix}\tag{6} H_\textrm{0L}:\frac{\mu_\textrm{T}}{\mu_\textrm{R}}\leq\theta_1\:vs\:H_\textrm{1L}:\frac{\mu_\textrm{T}}{\mu_\textrm{R}}>\theta_1\\ H_\textrm{0U}:\frac{\mu_\textrm{T}}{\mu_\textrm{R}}\geq\theta_2\:vs\:H_\textrm{1U}:\frac{\mu_\textrm{T}}{\mu_\textrm{R}}<\theta_2 \end{matrix}\]

If both $\small{p}$-values are $\small{\leq\alpha}$, the joint null hypothesis $\small{\left\{H_\text{0L},H_\text{0U}\right\}}$ is rejected and BE concluded.
If one $\small{p}$-value is $\small{>\alpha}$ and the other $\small{\leq\alpha}$, the outcome is inconclusive.
If both $\small{p}$-values are $\small{>\alpha}$, the joint null hypothesis is not rejected and inequivalence proven.

The two-sided $\small{1-2\,\alpha}$ confidence interval is assessed for inclusion in the acceptance range $\small{\left\{\theta_1,\theta_2\right\}}$.

\[H_0:\frac{\mu_\textrm{T}}{\mu_\textrm{R}}\not\subset\left\{\theta_1,\theta_2\right\}\:vs\:H_1:\theta_1<\frac{\mu_\textrm{T}}{\mu_\textrm{R}}<\theta_2\tag{7}\]

If the CI lies entirely within $\small{\left\{\theta_1,\theta_2\right\}}$, the null hypothesis $\small{H_0}$ is rejected and BE concluded.
If one confidence limit is outside $\small{\left\{\theta_1,\theta_2\right\}}$ and the other CL is within $\small{\left\{\theta_1,\theta_2\right\}}$, the outcome is inconclusive.
The same holds if CL_lower < $\small{\theta_1}$ and CL_upper > $\small{\theta_2}$.
If the CI lies entirely outside $\small{\left\{\theta_1,\theta_2\right\}}$, the null hypothesis is not rejected and inequivalence proven.

Evaluating our example for $\small{\left\{\theta_1,\theta_2\right\}=\left\{80\%,120\%\right\}}$ by $\small{(6)}$ we get $\small{p(\theta_0\geq\theta_1)=0.0160}$ and $\small{p(\theta_0\leq\theta_2)=0.0528}$. Since one of the $\small{p\textsf{-}}$values is $\small{>\alpha}$, the study would fail. Assessing it by $\small{(7)}$ we get a CI of 82.44 – 124.21%. The study would fail because the upper CL is > 120%.
Although the study failed, repeating it in a larger sample size (with higher power) would likely allow us to demonstrate BE, since the outcome was inconclusive.

If assessing the example by $\small{(7)}$ according to current standards (i.e., of $\small{\log_e\textsf{-}}$transformed data for $\small{\left\{\theta_1,\theta_2\right\}=}$ $\small{\{80\%,}$ $\small{125\%\}}$), we would get a 90% CI of 87.40 – 121.73% and the study would pass. The Times They Are a-Changin’.

Interlude 2

It is a misconception that a certain CI of a sample (i.e., a particular study) contains the – true (but unknown) – population mean $\small{\mu}$ with $\small{1-\alpha}$ probabilty. Let’s simulate some studies and evaluate them by $\small{(7)}$:

invisible(library(PowerTOST))
set.seed(123) # for reproducibility of simulations
mue      <- 1 # true population mean
CV       <- 0.25
studies  <- 100
x        <- sampleN.TOST(CV = CV, theta0 = mue, targetpower = 0.8, print = FALSE)
subjects <- x[["Sample size"]]
power    <- x[["Achieved power"]]
# simulate subjects within studies, lognormal distribution
samples  <- data.frame(study     = rep(1:studies, each = subjects * 2),
                       subject   = rep(rep(1:subjects, studies), each = 2),
                       period    = rep(rep(1:2, studies), 2),
                       sequence  = rep(c(rep(c("TR"), subjects),
                                         rep(c("RT"), subjects)), studies),
                       treatment = c(rep(c("T", "R"), subjects / 2),
                                     rep(c("R", "T"), subjects / 2)),
                       Y         = rlnorm(n = subjects * studies * 2,
                                          meanlog = log(mue) - 0.5 * log(CV^2 + 1),
                                          sdlog = sqrt(log(CV^2 + 1))))
facs     <- c("subject", "period", "treatment")
samples[facs] <- lapply(samples[facs], factor) # factorize the data
result   <- data.frame(study = 1:studies, PE = NA_real_,
                       lower = NA_real_, upper = NA_real_,
                       BE = FALSE, contain = TRUE)
grand.PE <- numeric(studies)
for (i in 1:studies) {
  temp           <- samples[samples$study == i, ]
  heretic        <- lm(log(Y) ~ period + subject + treatment, data = temp)
  result$PE[i]   <- 100 * exp(coef(heretic)[["treatmentT"]])
  result[i, 3:4] <- 100 * exp(confint(heretic, level = 0.90)["treatmentT", ])
  if (round(result[i, 3], 2) >= 80 & round(result[i, 4], 2) <= 125)
    result$BE[i] <- TRUE
  if (result$lower[i] > 100 * mue | result$upper[i] < 100 * mue) result$contain[i] <- FALSE
  grand.PE[i]    <- mean(result$PE[1:i]) # (cumulative) grand means
}
dev.new(width = 4.5, height = 4.5)
op       <- par(no.readonly = TRUE)
par(mar = c(3.05, 2.9, 1.4, 0.75), cex.axis = 0.9, mgp = c(2, 0.5, 0))
xlim     <- range(c(min(result$lower), 1e4 / min(result$lower),
                    max(result$upper), 1e4 / max(result$upper)))
plot(1:2, 100 * rep(mue, 2), type = "n", log = "x", xlab = "PE [90% CI]",
     ylab = "study  #", axes = FALSE,
     xlim = xlim, ylim = range(result$study))
abline(v = 100 * c(0.8, mue, 1.25), lty = c(2, 1, 2))
axis(1, at = c(125, pretty(xlim)),
     labels = sprintf("%.0f%%", c(125, pretty(xlim))))
axis(2, at = c(1, pretty(1:studies)[-1]), las = 1)
axis(3, at = 100 * mue, label = expression(mu))
box()
lines(grand.PE, 1:studies, lwd = 2)
for (i in 1:studies) {
  if (result$BE[i]) {       # pass
    clr <- "blue"
  } else {                  # fail
    if (result$contain[i]) {# mue within CI
      clr <- "magenta"
    } else {                # mue not in CI
      clr <- "red"
    }
  }
  lines(c(result$lower[i], result$upper[i]), rep(i, 2), col = clr)
  points(result$PE[i], i, pch = 16, cex = 0.6, col = clr)
}
par(op)

$Fig. 3 2×2×2 crossover studies (\small{\mu} = 100%, \small{CV} = 25%: \small{n} = 24 for ≥80% power).$

Fig. 3 2×2×2 crossover studies ($\small{\mu}$ = 100%, $\small{CV}$ = 25%: $\small{n}$ = 24 for ≥80% power).

In 7% of studies the population mean $\small{\mu}$ is not contained in the 90% CI (red lines). In other words, given the result of a single study we can never know where $\small{\mu}$ lies. Only the grand mean (mean of sample means $\small{\frac{1}{n}\sum_{i=1}^{i=n}\overline{x_i}}$) approaches $\small{\mu}$ for a large number of samples. After the 100^th study it is with 99.44% pretty close to $\small{\mu}$ (for geeks: The convergence is poor; if simulating 25,000 studies, it is 100.23%). However, nobody would repeat a – passing – study (blue lines) for such a rather uninteresting information, right?
This explains also why a particular study might fail by pure chance even if a formulation is equivalent (here 15% of studies; red or magenta lines). Such cases are related to the producer’s risk (Type II Error = 1 – power), which is for the given conditions 16.3%. On the other hand, it is also possible that a formulation which is not equivalent might pass. These cases are related to the patient’s risk (Type I Error).
For details see the articles about hypotheses, treatment effects, post hoc power, and sample size estimation. Science is a cruel mistress.

Let’s simulate a large number of studies and explore the probability of the Type I Error $\small{\alpha}$ (patient’s risk), the Type II Error $\small{\beta}$ (producer’s risk), and the power $\small{\pi}$ (chance to show BE, where $\small{\pi=1-\beta}$).
For an assumed $\small{CV_\text{intra}}$ of 25% ($\small{s^2\approx}$ 0.060625) and an assumed $\small{\mu}$ of 0.95 we require 28 subjects in a 2×2×2 crossover design to achieve at least 80% power. It must be mentioned that by the TOST Procedure $\small{(6)}$ or the CI inclusion approach $\small{(7)}$ the Type I Error is always strictly controlled (i.e., $\small{p(\text{TIE})\leq}$ $\small{\alpha}$).

invisible(library(PowerTOST))
set.seed(123456)
prep <- function(CV = CV, mue = mue, n = n) {
  # simulate subject data for a 2×2×2 crossover design
  x <- data.frame(subject   = rep(1L:n, each = 2),
                  period    = rep(1L:2L, n),
                  sequence  = c(rep("TR", n), rep("RT", n)),
                  treatment = c(rep(c("T", "R"), n / 2),
                                rep(c("R", "T"), n / 2)),
                  Y         = NA_real_)
  for (i in 1L:nrow(x)) {
    if (x$treatment[i] == "T") {
      x$Y[i] <- exp(1 + rnorm(n = 1, mean = log(mue), sd = CV2se(CV)))
    } else {
      x$Y[i] <- exp(1 + rnorm(n = 1, mean = 0, sd = CV2se(CV)))
    }
  }
  factors    <- c("subject", "period", "sequence", "treatment")
  x[factors] <- lapply(x[factors], factor) # required for lm()
  return(x)
}
mue      <- 0.95
CV       <- 0.25
design   <- "2x2x2"
target   <- 0.80
x        <- sampleN.TOST(CV = CV, theta0 = mue, targetpower = target,
                         design = design, print = FALSE)
n        <- x[["Sample size"]]
pwr.xact <- x[["Achieved power"]]
nsim     <- 10000L
study    <- data.frame(sim = 1:nsim, BE = FALSE, contain = TRUE)
op       <- options()
options(contrasts = c("contr.treatment", "contr.poly"))
for (i in 1L:nsim) {
  data  <- prep(CV, mue, n)
  # bogus nested model according to the guidelines!
  mudel <- lm(log(Y) ~ period + subject %in% sequence + treatment, data)
  CI    <- round(exp(confint(mudel, level = 0.9)["treatmentT", ]), 4)
  if (CI[1] >= 0.8 & CI[2] <= 1.25) study$BE[i]      <- TRUE
  if (CI[1] > mue  | CI[2] <  mue)  study$contain[i] <- FALSE
}
options(op)
aggregate <- data.frame(method = c("power.TOST()",
                                   "power.TOST.sim()",
                                   "script"),
                        details = c("exact (Owen’s Q)",
                                    "via assoc. statistics",
                                    "subject simulations"),
                        T1E    = c(power.TOST(CV = CV, theta0 = 1.25, n = n),
                                   power.TOST.sim(CV = CV, theta0 = 1.25, n = n,
                                                  nsim = nsim),
                                   (1 - sum(study$contain) / nsim) / 2),
                        T2E    = NA_real_,
                        power  = c(pwr.xact,
                                   power.TOST.sim(CV = CV, theta0 = mue, n = n,
                                                  nsim = nsim),
                                   sum(study$BE) / nsim))
aggregate[, 4]   <- 1 - aggregate[, 5]
for (i in 3:5) { # cosmetics
  aggregate[, i] <- sprintf("%.4f%%", 100 * aggregate[, i])
}
names(aggregate)[3:5] <- c("  T1E", "   T2E", "  power")
print(aggregate, row.names = FALSE, right = FALSE)

#  method           details                 T1E      T2E     power 
#  power.TOST()     exact (Owen’s Q)      5.0000% 19.2561% 80.7439%
#  power.TOST.sim() via assoc. statistics 4.5800% 19.0200% 80.9800%
#  script           subject simulations   4.9800% 19.2500% 80.7500%

In simulating 10,000 studies, we get a close match to the exact values – both by the parameter’s associated statistics (i.e., $\small{\log_\text{e}(\mu)}$ via the normal distribution and $\small{s^2}$ via its associated $\small{\chi^2\textsf{-}}$distribution with $\small{n-2}$ degrees of freedom), as well as by subject simulations.
Of couse, the exact $\small{p(\text{TIE})\approx 0.049999629914<\alpha}$. q.e.d.

As usual in such simulations, the convergence (especially of the Type I Error) is poor. For a closer match to the exact values we would need at least one million simulations. In the Monte Carlo method values approach the exact one asymptotically. Here the exact and empiric Type I Errors for 10⁶ simulations obtained by a slightly modified script:

#  method           details                  T1E     bias
#  power.TOST()     exact (Owen’s Q)      4.99996%    –
#  power.TOST.sim() via assoc. statistics 5.00380% +0.0767%
#  script           subject simulations   4.99130% -0.1733%

$Fig. 3a 2×2×2 crossover studies (\small{\mu} = 0.95, \small{CV} = 25%, \small{\pi\geq 80\%}: \small{n} = 28).$

Fig. 3a 2×2×2 crossover studies ($\small{\mu}$ = 0.95, $\small{CV}$ = 25%, $\small{\pi\geq 80\%}$: $\small{n}$ = 28).

Fig. 3b Bias of the empiric Type I Error to exact one (≈ 0.049999629914).

Fig. 3c Enlarged section of Fig. 3b.

#  sim. studies  bias (%)
#         1,000 +12.00083
#        10,000  -8.39932
#       100,000  -1.15927
#     1,000,000  +0.07674
#    10,000,000  -0.02146
#   100,000,000  -0.00596

Of course, this is of academic interest only because the exact result is readily available in the blink of an eye.

At a hearing in 1986 the FDA confirmed that $\small{(6)}$ or $\small{(7)}$ of untransformed data should be used with $\small{\Delta=}$ $\small{20\%}$. If clinically relevant, tighter limits ($\small{\Delta=10\%}$) might be needed.⁶⁶

The first German guideline was drafted by the International Association for Pharmaceutical Technology (Arbeitsgemeinschaft für Pharmazeutische Verfahrenstechnik) in 1985.⁶⁷ It was presented and discussed in 1987.⁶⁸ ⁶⁹ ⁷⁰ In the the same year, the first guideline of the Nordic Council on Medicines was published in cooperation with the agencies of Denmark, Finland, Iceland, Norway and Sweden.⁷¹

In 1988 wider acceptance limits of 70 – 130% were proposed for $\small{C_\text{max}}$ due to its inherent high variability⁷² (as a one-point metric practically always larger than the one of the integrated metric $\small{AUC}$).

The Australian draft guideline was published in 1988.⁷³ It was the first covering not only the design and evaluation but also validation of bioanalytical methods. The model with effects period, subject, treatment^{25 52} was recommended and a test for sequence-effects was not considered necessary. The problematic conversion of differences to percentages was acknowledged and Fieller’s CI^26
27 mentioned. Kudos to both!

In 1989 a loose-leaf collection was started.⁷⁴ It contained raw-data of generic drugs marketed in Germany, the evaluation provided by companies, as well as results recalculated by the ZL (Central Laboratory of German Pharmacists). Including the 6^th supplement of 1996 it contained more than 2,000 pages… It was an indispensible resource for planning new studies and also showed the ‘journey’ of dossiers (i.e., the same study being used by different companies).

The BioInternational conferences co-organized by Henning Blume and Kamal K. Midha (Toronto 1989, Bad Homburg 1992, Munich 1994, Tokio 1996, London 1999, 2003, 2005, 2008) made valuable contributions to the advancement of testing for bioequivalence. The first dealt with the $\small{\log_{e}\textsf{-}}$transformation of data and the definition of Highly Variable Drugs (HVDs).⁷⁵ There was a poll among the participants about the $\small{\log_{e}\textsf{-}}$transformation of data. Outcome: ⅓ never, ⅓ always, ⅓ case by case (i.e., perform both analyses and report the one with narrower CI ‘because it fits the data better’). Let’s be silent about the last team.⁷⁶ HVDs were defined as drugs with intra-subject variabilities of more than 30% but problems might be evident already with 25%.

top of section ↩︎ previous section ↩︎

The Boring (?) 1990s

The original acceptance range was symmetrical around 100%. In $\small{\log_{e}\textsf{-}}$scale it should be symmetrical around $\small{0}$ (because $\small{\log_{e}1=0}$). What happens to our $\small{\Delta}$, which should still be 20%? Due to the positive skewness of the lognormal distribution a lively discussion started after early publications proposing 80 – 125%.^{25 52} Keeping 80 – 120% would have been flawed because the maximum power should be obtained at $\small{\mu_\text{T}/\mu_\text{R}=1}$ for \[\exp\left((\log_{e}\theta_1+\log_{e}\theta_2)/2\right),\tag{8}\] which works only if $\small{\theta_2=\theta_1^{-1}}$ or $\small{\theta_1=\theta_2^{-1}}$. Keeping the original limits, maximum power would be obtained at $\small{\mu_\text{T}/\mu_\text{R}=}$ $\small{\exp((\log_{e}0.8+\log_{e}1.2)/2)}$ $\small{\approx0.979796}$.

Fig. 4 Power for a 2×2×2 design and limits 0.80 – 1.20.
Note that with a multiplicative model the power curve is asymmetric
and shifted to the left.

There were three parties (all agreed that the acceptance range should be symmetrical in $\small{\log_{e}\textsf{-}}$scale and consequently asymmetrical when back-transformed). These were their arguments and suggestions:

The width of the acceptance range was 40% and we have empiric evidence that the concept of BE ‘worked’ – let’s be conservative and keep it.

\[\left\{\theta_1,\theta_2\right\}=81.98-121.98\%\tag{9}\]

Since that’s a new method, we don’t want to face safety issues with a higher limit. Furthermore, a more restrictive lower limit prevents issues with insufficient efficacy.

\[\left\{\theta_1,\theta_2\right\}=\left\{100/(1+\Delta),100\,(1+\Delta)\right\}=8\dot{3}.33-120\%\tag{10}\]

80% as the lower limit served us well in the past. Hence, 125% is the way to go because it is simply the reciprocal of the lower limit and the coverage probability in the log-domain is the same like the one we had. Furthermore, these are nice numbers.

\[\left\{\theta_1,\theta_2\right\}=\left\{100\,(1-\Delta),100/(1-\Delta)\right\}=80-125\%\tag{11}\]

The 90% CI inclusion approach $\small{(7)}$ based on $\small{\log_{e}\textsf{-}}$transformed data with acceptance limits of 80.00 – 125.00% $\small{(5)}$ was the winner.

$Fig. 5 Power for a 2×2×2 design and limits 0.80 – 1.25. Note the symmetry: power for any \small{1/\theta=\theta}.$

Fig. 5 Power for a 2×2×2 design and limits 0.80 – 1.25.
Note the symmetry: power for any $\small{1/\theta=\theta}$.

Why are we using a 90% CI – and not a 95% CI like in phase III? In the worst case in a particular patient the bioavailability can be

either too low, i.e., $\small{p(\text{BA}<\phantom{1}80\%)>5\%}$

or too high, i.e., $\small{p(\text{BA}>125\%)>5\%}$

but evidently not at the same time. Hence, the 90% CI controls the risk for the population of patients. Therefore, if a study passes, the risk for patients does still not exceed 5%. Note that at the BE limits $\small{\left\{\theta_1,\theta_2\right\}}$ power, i.e., the chance to pass, is 5%. Therefore, the patient’s risk (type I error) is controlled.

Fig. 6 Top: 5% risk for patients with low BA
Middle: 5% risk for patients with high BA
Bottom: 90% CI for the population of patients unveiled

First sample size tables for the multiplicative model with the acceptance range 80 – 125% were published⁷⁷ and extended for narrower ($\small{\Delta=10\%}$: 90.00 – 111.11%) and wider ($\small{\Delta=30\%}$: 70.00 – 142.86%) acceptance ranges.⁷⁸ The nonparametric method was improved taking period-effects into account.⁷⁹ ⁸⁰ Drug-drug and food-interaction studies should be assessed for equivalence.⁸¹ The general applicability of average BE was challenged and the concept of individual and population bioequivalence outlined.⁸² ⁸³ ⁸⁴ The first textbook dealing exclusively with BA/BE was published.⁸⁵

This was also the decade of updated and new guidelines. A European draft guidance was published in 1990;⁸⁶ the final guideline was published in December 1991 and came into force in June 1992.⁸⁷ The 90% CI inclusion approach of $\small{\log_{e}\textsf{-}}$transformed data with an acceptance range of 80 – 125% was recommended and for NTIDs the acceptance range may need to be tightened. Due to its inherent higher variability a wider acceptance range may be acceptable for $\small{C_\text{max}}$. If inevitable and clinically acceptable, a wider acceptance range may also be used for $\small{AUC}$. Only if clinically relevant, a nonparametric analysis of $\small{t_\text{max}}$ was recommended.

An in vivo study was not required if the new formulation is

to be parenterally administered as a solution and contains the same API(s) and excipients in the same concentrations as the reference or
is a liquid oral form in solution (elixir, syrup, etc.) containing the API(s) in the same concentration and form as the reference, not containing excipients that may significantly affect gastric passage or absorption of the active substance.

Similar statements about solutions were given in all later guidelines. The second lead to application of the Biopharmaceutic Classification System (BCS).⁸⁸ More about that further down.

“The almost classical 1977 FDA notice […] defined bioavailability as the rate and extent to which the active drug ingredient of therapeutic moiety is absorbed from a drug product and becomes available at the site of action.²⁰ However, in the majority of cases substances are intended to exhibit a systemic therapeutic effect, and a more practical definition can be given, taking into account that the substance in the general circulation is in exchange with the substance at the site of action. Therefore, the European 1991 guidance on bioavailability and bioequivalence⁸⁷ gave the following definition: Bioavailability is understood as to be the extent and rate to which the a substance or its therapeutic moiety is delivered from the pharmaceutical form into the general circulation.
Volker W. Steinijans and Dieter Hauschke (1993)⁸⁹

In July 1992 a guidance of the FDA was published.⁹⁰ An ANOVA of $\small{\log_{e}\textsf{-}}$transformed data was recommended and the nested subject(sequence) term in the statistical model entered the scene.
It must be mentioned that in comparative BA studies subjects are usually uniquely coded. Hence, the term subject(sequence) is a bogus one⁹¹ and could be replaced by the simple subject as well (see below for an example). Alas, this model was implemented in all global guidelines ever since. If you understand why, let me know.

In the same year the Canadian guidance for Immediate Release (IR) formulations was published.⁹² To that time is was the most extensive one because it gave not only the method of evaluation, but information about the study design, sample size, ethics, bioanalytics, etc. It differed from the others in the relaxed requirement for $\small{C_\text{max}}$, where only the $\small{\text{T}/\text{R}\textsf{-}}$ratio has to lie within 80 – 125% (instead of its CI). The guidance for MR formulations followed in 1996.⁹³

In 1998 the World Health Organization published its first guideline,⁹⁴ which was similar to the European one.

Table IV shows the result of the example evaluated by various methods. \[\small{\begin{array}{lcccc} \textsf{Table IV}\phantom{0}\\ \phantom{0}\text{Method} & \text{Model} & \text{PE} & \text{power},p,\text{CI, etc.} & \text{BE?}\\\hline \text{80/20 Rule} & \text{additive} & - & 46.40<80\% & \text{fail}\\ t\text{-test} & \text{additive} & +2.417\;(103.32\%) & 0.7193\geq0.05 & \text{pass}\\ \text{TOST} & \text{additive} & +2.417\;(103.32\%) & 0.0160\leq0.05,\,0.0528>0.05 & \text{fail}\\ \text{95% CI} & \text{additive} & +2.417\;(103.32\%) & -12.777\,,+17.611\;(82.44-124.21\%) & \text{fail}\\ \text{Fieller} & \text{additive} & 103.32\% & 84.84-125.65\% & \text{fail}\\ \text{Westlake} & \text{additive} & \pm0.000\;(100.00\%) & \pm2.944\;(\pm21.80\%) & \text{fail}\\\hline \text{80/20 Rule} & \text{multiplicative} & - & 72.90<80\% & \text{fail}\\ \text{75/75 Rule} & \text{(multiplicative)} & - & 9/12=75\% & \text{pass}\\ t\text{-test} & \text{multiplicative} & 103.14\% & 0.7317\geq0.05 & \text{pass}\\ \text{TOST} & \text{multiplicative} & 103.14\% & 0.0097\leq0.05,\,0.0309\leq0.05 & \text{pass}\\ {\color{Blue} {90\%\,\text{CI}}} & {\color{Blue} {\text{multiplicative}}} & {\color{Blue} {103.14\%}} & {\color{Blue} {87.40-121.73\%}} & {\color{Blue} {\text{pass}}}\\ \text{Westlake} & \text{multiplicative} & 100.00\% & \pm18.09\% & \text{pass}\\ \text{75/75 Rule} & \text{multiplicative} & - & 75\%\subset \pm25\% & \text{pass}\\\hline \end{array}}\] In the additive model the acceptance range was 80 – 120%, whereas in the multiplicative model it is 80 – 125%. Since in the former differences are assessed, the wrong percentages are given in brackets.

At the time being only the 90% CI inclusion approach is globally accepted. Our example in R again:

example       <- data.frame(subject   = rep(1:12, each = 2),
                            sequence  = c("RT", "RT", "TR", "TR", "RT",
                                          "RT", "TR", "TR", "TR", "TR",
                                          "RT", "RT", "RT", "RT", "TR",
                                          "TR", "TR", "TR", "RT", "RT",
                                          "RT","RT",  "TR", "TR"),
                            treatment = c("R", "T", "T", "R", "R", "T", "T", "R",
                                          "T", "R", "R", "T", "R", "T", "T", "R",
                                          "T", "R", "R", "T", "R", "T", "T", "R"),
                            period    = rep(1:2, 12),
                            Y         = c(81, 71, 61, 65, 94, 80, 66, 74,
                                          94, 54, 63, 97, 85, 70, 76, 90,
                                          54, 53, 56, 99, 90, 83, 51, 68))
facs          <- c("subject", "sequence", "treatment", "period")
example[facs] <- lapply(example[facs], factor) # factorize the data
txt           <- paste("nested model : period, subject(sequence), treatment",
                       "\nsimple model : period, subject, sequence, treatment",
                       "\nheretic model: period, subject, treatment\n\n")
result        <- data.frame(model = c("nested", "simple", "heretic"),
                            PE = NA, lower = NA, upper = NA, BE = "fail", na = 0)
for (i in 1:3) {
  if (result$model[i] == "nested") { # bogus nested model (guidelines)
    nested         <- lm(log(Y) ~ period +
                                  subject %in% sequence +
                                  treatment, data = example)
    result$PE[i]   <- 100 * exp(coef(nested)[["treatmentT"]])
    result[i, 3:4] <- 100 * exp(confint(nested, level = 0.90)["treatmentT", ])
    result[i, 6]   <- sum(is.na(coef(nested)))
  }
  if (result$model[i] == "simple") { # simple model (subjects are uniquely coded)
    simple         <- lm(log(Y) ~ period +
                                  subject +
                                  sequence +
                                  treatment, data = example)
    result$PE[i]   <- 100 * exp(coef(simple)[["treatmentT"]])
    result[i, 3:4] <- 100 * exp(confint(simple, level = 0.90)["treatmentT", ])
    result[i, 6]   <- sum(is.na(coef(simple)))
  }
  if (result$model[i] == "heretic") { # heretic model (without sequence)
    heretic        <- lm(log(Y) ~ period +
                                  subject +
                                  treatment, data = example)
    result$PE[i]   <- 100 * exp(coef(heretic)[["treatmentT"]])
    result[i, 3:4] <- 100 * exp(confint(heretic, level = 0.90)["treatmentT", ])
    result[i, 6]   <- sum(is.na(coef(heretic)))
  }
  # rounding acc. to guidelines
  if (round(result[i, 3], 2) >= 80 & round(result[i, 4], 2) <= 125)
    result$BE[i] <- "pass"
}
checks        <- data.frame(comparison = c("simple", "heretic"),
                            lower = "different", upper = "different")
for (i in 2:3) {
  if (isTRUE(all.equal(result$lower[i], result$lower[1])))
    checks$lower[i-1] <- "identical"
  if (isTRUE(all.equal(result$upper[i], result$upper[1])))
    checks$upper[i-1] <- "identical"
}
# cosmetics
names(checks) <- c("Comparison vs nested", "lower CL", "upper CL")
result$PE     <- sprintf("%6.2f%%", result$PE)
result$lower  <- sprintf("%6.2f%%", result$lower)
result$upper  <- sprintf("%6.2f%%", result$upper)
names(result)[c(3:4, 6)] <- c("lower CL", "upper CL", "NE")
cat(txt); print(result, row.names = FALSE); print(checks, row.names = FALSE)

# nested model : period, subject(sequence), treatment 
# simple model : period, subject, sequence, treatment 
# heretic model: period, subject, treatment
# 
#    model      PE lower CL upper CL   BE NE
#   nested 103.14%   87.40%  121.73% pass 13
#   simple 103.14%   87.40%  121.73% pass  1
#  heretic 103.14%   87.40%  121.73% pass  0
#  Comparison vs nested  lower CL  upper CL
#                simple identical identical
#               heretic identical identical

As already outlined above, the nested model recommended in all [sic] guidelines is over-specified because subjects are uniquely coded.

If you don’t believe that the results of all models are identical, try it with any of your studies.

In the example we get 13 not estimable (aliased) effects. Correct, because we are asking for something the data cannot provide.⁹¹ In the simple model only one effect cannot be estimated. However, even sequence can be removed from the model. I call it heretic because regulators will grill you if you are using it. It was proposed by Westlake^25
52 and I employed it in hundreds (‼) of studies and some cases are published.⁹⁵

A ‘Positive List’ was published by the German regulatory authority, i.e., for 90 drugs BE was not required.⁹⁶ In order to comply with the European Note for Guidance of 2001⁹⁷ it had to be removed by the BfArM.

The FDA guidance for ‘Scale-Up and Postapproval Changes’ (SUPAC)⁹⁸ ⁹⁹ defined three ‘Levels’ of changes:

Those that are unlikely to have any detectable impact on formulation quality and performance.
Those that could have a significant impact on formulation quality and performance. Tests and filing documentation for a Level 2 change vary depending on three factors: therapeutic range, solubility, and permeability. Therapeutic range is defined as either narrow or non-narrow. […] Drug solubility and drug permeability are defined as either low or high. Solubility is calculated based on the minimum concentration of drug (mg/mL), in the largest dosage strength, determined in the physiological pH range (pH 1 to 8) and temperature (37 ±0.5 ℃). High solubility drugs are those with a dose / solubility volume of less than or equal to 250 mL. Permeability P_e (cm/s) is defined as the effective human jejunal wall permeability of a drug and includes an apparent resistance to mass transport to the intestinal membrane. High permeability drugs are generally those with an extent of absorption greater than 90% in the absence of documented instability in the gastrointestinal tract, or those whose permeability attributes have been determined experimentally.
Those that are likely to have a significant impact on formulation quality and performance. Tests and filing documentation vary depending on the following three factors: therapeutic range, solubility, and permeability.

Under certain conditions of Level 2, demonstration of in vitro similarity by $\small{f_2\geq 50\%}$¹⁰⁰ in the application / compendial medium at 15, 30, 45, 60 and 120 minutes (or until an asymptote is reached) of at least 12 units is sufficient.

\[f_2=50\,\log_{10}\left\{100\,\sqrt{1+\frac{1}{n}\sum_{i=1}^{i=n}(\text{R}_i-\text{T}_i)^2}\right\}\small{\textsf{,}}\tag{12}\]

where $\small{\text{R}_i}$ and $\small{\text{T}_i}$ are the cumulative percent dissolved at $\small{1\ldots\ n}$ time points of $\small{\text{R}}$ and $\small{\text{T}}$, respectively.
For Level 3 changes in vivo testing (BE) is mandatory.

It must be mentioned that comparing formulations by $\small{f_2}$ can be problematic, especially if the shapes of dissolution curves are different and/or if they intersect. $\small{f_2}$ is not a statistic and, therefore, it is impossible to evaluate false positive and negative rates of decisions for approval of drug products based on it.¹⁰¹

Two (of five) sessions of the BioInternational ’92 conference in Bad Homburg dealt with BE of Highly Variable Drugs.¹⁰² ¹⁰³ Various approaches have been discussed: Multiple dose instead of single dose studies, metabolite instead of the parent compound, stable isotope techniques,¹⁰⁴ add-on designs, and – for the first time – replicate designs.

Although the BioInternational 2 in Munich 1994 was with over 600 participants the largest in the series, no substantial progress for HVD(P)s was achieved.¹⁰⁵ Following a suggestion¹⁰⁶ at a joint AAPS/FDA workshop in 1995 widening the conventional acceptance limits of 80.00 – 125.00% was considered.¹⁰⁷

“For some highly variable drugs and drug products, the bioequivalence standard should be modified by changing the BE limits while maintaining the current confidence interval at 90%. […] the bioequivalence limits should be determined based in part upon the intrasubject variability for the reference product.
Shah et al. (1996)¹⁰⁷

A hot topic ever since… Why are we discussing it for 36 (‼) years (since the first BioInternational conference)? Is it really that complicated¹⁰⁸ or are we too stupid?

Studies in steady-state were proposed as an option for HVD(P)s in a European draft guideline¹⁰⁹ in order to reduce variability, but it was removed from the final version of 2001.⁹⁷

Validation of bioanalytical methods¹¹⁰ ¹¹¹ ¹¹² ¹¹³ was partly covered in Australia and Canada. However, no specific guideline existed. A series of conferences (informally known as ‘Crystal City’) was initiated in 1990.¹¹⁴ Procedures stated in the conference report¹¹⁵ were discussed at the BioInternational 2 in Munich 1994 and quickly adopted by bioanalytical sites.

In 1996 the WHO initiated the ‘International Comparator Product System for Pharmaceuticals’ to establish a ‘Global Comparator’, which could be used in countries where the innovator’s product is not – or no more – marketed. As the product may have been changed (not necessarily in all countries), the innovators were contacted and asked which product is currently closest to the one that led to the original authorization. These letters were ignored. There was not even an answer like »We have received your request but prefer rather not to reply because the information is confidential.« In light of this, the WHO requested that the competent authorities of the countries inquire once more. Unfortunately, this request has been largely ignored by the majority of innovators.¹¹⁶ The list of international comparator products was first published in June 1999 and is updated periodically.¹¹⁷
An innovator showed an example of a product which has undergone more than twenty (‼) changes, with the product closest to the original being marketed only in Nigeria…¹¹⁸

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21^st century

Poland happily adopted Germany’s ‘Positive List’⁹⁶ only when it wanted to join the European Union to learn that in the meantime Germany abandoned it to comply with the 2001 guideline.⁹⁷
A positive list of 19 drugs existed in The Netherlands for »strict national market authorisation«.¹¹⁹ It must have been a schizophrenic situation for assessors of the MEB: In the morning a dossier for national MA of IR paracetamol without any in vivo comparison → ✔ accepted. In the afternoon another dossier of the same product in the course of a European submission. A comparative BA study performed, but 90% CI 80.00–125.01% → ✘ rejected. Outright bizarre.
Denmark required that the 90% CI had to include 100% (i.e., that there is no significant treatment effect).¹²⁰ Bizarre as well. For details see the examples in this article.

In February 2005 the FDA published the Electronic Orange Book (EOB), which is updated daily. It can be searched by: Proprietary name, active ingredient, applicant (company), application number, dosage form, route of administration, patent number. It gives also a list of newly added or delisted patents.

The series of ‘Crystal City’ meetings continued.¹²¹ ¹²² Incurred sample reanalysis (ISR) was proposed¹²² and details subsequently outlined.¹²³ The first bioanalytical method validation guidance was published by the FDA in 2001 and revised in 2018.¹²⁴ ¹²⁵ Before the EMA’s draft guideline was published in 2009,¹²⁶ some European inspectors raised an eyebrow if sites worked according to a ‘foreign’ (i.e., the FDA’s) guidance.

“The validation of bioanalytical methods and the analysis of study samples should be performed in accordance with the principles of Good Laboratory Practice (GLP). However, as human bioanalytical studies fall outside of the scope of GLP […], the sites conducting the human studies are not required to be monitored as part of a national GLP compliance programme.
EMEA (2009)¹²⁶

Well roared, lions! My CRO (in Austria) was GLP-certified since 1991, although we performed only phase I studies. In other countries (e.g., Spain), this was not possible. In Germany GLP is subject to state law. Hence, it was possible to get certified in one federal state but not in another… However, this ‘issue’ was resolved with the final guideline published in 2011¹²⁷ and the ICH M10 guideline of 2022,¹²⁸ ¹²⁹ superseding all local guidelines.

In June 2010 the FDA started to publish Product-Specific Guidances (PSGs).¹³⁰ They are available online (with November 19, 2024 an amazing 2,252) and can be searched by active ingredient or RLD. Many PSGs remain drafts for a long time. For example, of the 138 PSGs starting with the letter P, only five (‼) are final and some are for 14 years still in draft state.

The EMA requires for prolonged and multiphasic release products both a single dose study and a study in steady state.¹³¹ The steady state study can be waived if there is no ‘risk’ of accumulation ($\small{AUC_{0-\tau}>90\%AUC_{0-\infty}}$, where $\small{\tau}$ is the intended dosing interval). However, for prolonged release products this option is rarely – if ever – possible… Different PK metrics to assess the minimum concentration in steady-state are required for originators and generic companies. The former have to assess the minimum concentration within the dosing interval $\small{(C_\text{ss,min})}$, whereas the latter have to assess the minimum concentration at the end of the dosing interval $\small{(C_{\text{ss}\,,\tau})}$. If there is a lag-time, the latter is more difficult due to its higher variability.¹³² Why double standards?
For prolonged release products with no ‘risk’ of accumulation and multiphasic release products the cut-off times for partial $\small{AUC\textsf{s}}$ have to be pre-specified based on PK, which is a rather difficult feat. Furthermore, BE has not only to be demonstrated for $\small{AUC_{0-\text{t}}}$ and all partial $\small{AUC\textsf{s}}$ but also for $\small{C_\text{max}}$ in each of the sections. This gives for one cut-off time $\small{\text{tc}}$ already a whopping six PK metrics ($\small{AUC_{0-\text{t}}}$, $\small{AUC_{0-\infty}}$, $\small{AUC_{0-{\text{tc}}}}$, $\small{AUC_{\text{tc}-\text{t}}}$, $\small{C_{\text{max,t}\leq \text{tc}}}$, $\small{C_{\text{max,t}>\text{tc}}}$). At least reference-scaling (see below) is acceptable for all PK metrics – except for $\small{AUC_{0-\text{t}}}$ and $\small{AUC_{0-\infty}}$.¹³⁰ That’s different to the few PSGs of the FDA (e.g., methylphenidate, zolpidem), where the cut-off times are based on PD and – apart from the partial $\small{AUC\textsf{s}}$ and $\small{AUC_{0-\infty}}$ – only the global $\small{C_\text{max}}$ is required.

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PBE/IBE

Whereas Average Bioequivalence (ABE) is bijective (if $\small{T}$ is equivalent to $\small{R}$, $\small{R}$ is also equivalent to $\small{T}$), this holds in all variants of Scaled Average Bioequivalence (SABE) if and only if $\small{CV_\text{wT}=CV_\text{wR}}$. Therefore, switching from $\small{R}$ to $\small{T}$ is tolerable if $\small{CV_\text{wT}<CV_\text{wR}}$ but in such a case switching from $\small{T}$ to $\small{R}$ might be problematic.

After a wealth of – controversal – publications in the 1990s,^{82 83 84} ¹³³ ¹³⁴ ¹³⁵ ¹³⁶ ¹³⁷ ¹³⁸ ¹³⁹ ¹⁴⁰ ¹⁴¹ the FDA introduced two new concepts as alternatives to ABE, namely Population Bioequivalence (PBE) and Individual Bioequivalence (IBE).¹⁴² ABE focuses only on the comparison of population averages of the PK metrics and not the variances of formulations. It does also not assess a subject-by-formulation interaction variance, that is, the variation in the average $\small{\text{T}}$ and $\small{\text{R}}$ difference among individuals. In contrast, PBE and IBE include comparisons of both averages and variances of PK metrics. The PBE approach assesses total variability of the PK metrics in the population. The IBE approach assesses within-subject variability for the $\small{\text{T}}$ and $\small{\text{R}}$ formulations, as well as the subject-by-formulation interaction.

Demonstrated PBE would support ‘Prescribability’ (i.e., a drug naïve patient could start treatment), whereas IBE support ‘Switchability’ (i.e., a patient could switch formulations during treatment).¹⁴¹ Contrary to ABE, both PBE and IBE require studies in a full replicate design, which means that both $\small{\text{T}}$ and $\small{\text{R}}$ are administered twice. The acceptance limits for ABE were kept at 80.00 – 125.00% but for the others scaling to the variability of the reference was possible. That would mean an incentive for test formulations with a lower variability than the one of the reference but a penalty for ones with a higher variability.

However, the underlying statistical concepts were not trivial and the result practically incomprehensible for non-statisticians. Furthermore, both approaches had a discontinuity (when moving from constant- to reference-scaling), which lead to an inflated type I error (patient’s risk) of approximately 6.5%.^{137 139 142} ¹⁴³ ¹⁴⁴
PBE/IBE faced criticism, e.g.,

responses [to the guidance] were still doubt-filled as to whether the new bioequivalence criteria really provided added value compared to average bioequivalence¹⁴⁵

and was regarded a

‘theoretical’ solution to a ‘thoretical’ problem¹⁴⁶ ¹⁴⁷

leading to its omission from a subsequent guidance,¹⁴⁸ and a return to conventional ABE.¹⁴⁹

“Average bioequivalence should suffice based upon grounds of ‘practicality, plausibility, historical adequacy, and purpose’ and ‘because we have better things to do.’ […] ‘Statisticians have a bad track record in bioequivalence, […] the literature is full of ludicrous recommendations from statisticians, […] regulatory recommendations (of dubious validity) have been hastily implemented, and practical realities have been ignored’.
Stephen Senn (2000)¹⁵⁰
“Individual bioequivalence is a promising, clinically relevant method that should theoretically provide further confidence to clinicians and patients that generic drug products are indeed equivalent in an individual patient.
Even today, considering the studies summarized and analyzed by the FDA, the data is inadequate to validate the theoretical approach and provide confidence to the scientific community that the methodology required and the expense entailed are justified.
At this time, individual bioequivalence still remains a theoretical solution to solve a theoretical clinical problem. We have no evidence that we have a clinical problem, either a safety or an efficacy issue, and we have no evidence that if we have the problem that individual bioequivalence will solve the problem.
Leslie Z. Benet (2001)¹⁴⁶

I remember a Dutch regulator standing up in the BioInternational conference in London 2003, saying:

I’m glad that PBE and IBE are dead. I never understood them.

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SABE

Another possibility to deal with Highy Variable Drug Products – requiring extremely large sample sizes in ABE – is the concept of Scaled Average Bioequivalence (SABE). It was a topic at meetings of the FDA Advisory Committee for Pharmaceutical Science (7 May 1997, 23 September 1999, 14 April 2004, 10 October 2006). In a meeting of the Therapeutic Products Directorate (26–27 June 2003), Health Canada stated:

We don’t see a problem, our database search showed that even HVDPs comply with the usual BE criteria.

However, this observation was based on its requirement that only the PE of $\small{C_\text{max}}$ has to lie within 80.0–125.0%.⁹³

Benet proposed in a keynote at the BioInternational 1994 that innovators should perform replicate studies as part of the new drug application and provide information on intra- and inter-subject measures of extent and rate of BA in the PK section of the package insert.¹⁰⁵ SABE was also discussed at the BioInternational 2005.¹⁵¹

The EMA published a concept paper in 2006, containing valuable points for discussion.¹⁵²

What are the best methods to provide evidence that a medicinal product is a HVDP?
Describe different approaches to bioequivalence of HVDP, with benefits and drawbacks for regulatory purposes.
For the SABE concept:
- Define the recommended study designs.
- Define the acceptance range for this new approach.
- Suggest the recommended statistical and computational analyses, including the estimation of the within-subject variances of the two formulations and the determination of BE. A technical appendix will describe the recommended computational methods.
- Decide whether any additional constraints are necessary.
- Decide what to do if the within-subject variance ratio shows that the test product is more variable than the reference product.
- Decide how to define and how to handle outliers with this approach.

The concept paper was deleted from the EMA’s website in October 2007 – an unprecedented case…

Application of SABE was not limited to a certain PK metric. Furthermore, a comparison of $\small{s_{\text{wT}}^{2}}$ with $\small{s_{\text{wR}}^{2}}$ would require a full replicate design.

“Who controls the past controls the future:
who controls the present controls the past.
George Orwell (Nineteen Eighty-Four)

SABE was introduced 2010 first by the EMA,¹⁵³ shortly after by the FDA,¹⁵⁴ ¹⁵⁵ in 2017 by the WHO,¹⁵⁶ and in 2018 by Health Canada.¹⁵⁷

Terminology:

A Highy Variable Drug (HVD) shows a within-subject Coefficient of Variation of the Reference ($\small{CV_\text{wR}}$) > 30% if administered as a solution in a replicate design. The high variability is an intrinsic property of the drug (absorption, permeation, clearance – in any combination).
A Highy Variable Drug Product (HVDP) shows a $\small{CV_\text{wR}}$ > 30% in a replicate design.¹⁵⁸

The concept of SABE is based on the following considerations:

HVD(P)s are safe and efficacious despite their high variability because:
1. They have a wide therapeutic index (i.e., a flat dose-response curve). Consequently, even substantial changes in concentrations have only a limited impact on safety and efficacy.
  If they would have a narrow therapeutic index, adverse effects (due to high concentrations) and lacking effects (due to low concentrations) would have been observed in phase II (or in phase III at the latest) and therefore, the originator’s product should not have been approved in the first place.¹⁵⁹
2. Once approved, the product has a documented safety / efficacy record in phase IV and in clinical practice. If problems would became evident, the product would have been taken off the market.
Given that, the conventional ‘clinically relevant difference’ $\small{\Delta=20\%}$ in ABE is considered overly conservative and hence, requires large sample sizes.
Thus, a more relaxed $\small{\Delta>20\%}$ was proposed. A natural approach is to scale (expand / widen) the limits based on the within-subject variability of the reference product $\small{\sigma_\text{wR}}$.¹⁶⁰

The conventional model of ABE by $\small{(7)}$ is modified in SABE to \[H_0:\;\frac{\mu_\text{T}}{\mu_\text{R}}\Big{/}\sigma_\text{wR}\not\subset\left\{\theta_{\text{s}_1},\theta_{\text{s}_2}\right\}\;vs\;H_1:\;\theta_{\text{s}_1}<\frac{\mu_\text{T}}{\mu_\text{R}}\Big{/}\sigma_\text{wR}<\theta_{\text{s}_2},\tag{13}\] where $\small{\sigma_\text{wR}}$ is the standard deviation of the reference. The scaled limits $\small{\left\{\theta_{\text{s}_1},\theta_{\text{s}_2}\right\}}$ of the acceptance range depend on conditions given by the agency.

Reference-Scaled Average Bioequivalence (RSABE)¹⁶¹ is recommended by the FDA and China’s CDE. Average Bioequivalence with Expanding Limits (ABEL)¹⁶² is another variant of SABE and recommended in all other jurisdictions. In order to apply the methods following conditions have to be fulfilled:

The study has to be performed in a replicate design, i.e., at least the reference product has to be administered twice.
The observed within-subject variability of the reference product has to be high (in RSABE $\small{s_\text{wR}\geq 0.294}$¹⁶³ and in ABEL $\small{CV_\text{wR}>30\%}$).
Agencies are only interested in the variability of the reference product, although for the applicant the one of the test is ‘nice to know’ as well.
ABEL only:
1. A clinical justification must be given that the expanded limits will not impact safety / efficacy.
2. There is an ‘upper cap’ of scaling ($\small{uc=50\%}$, except for Health Canada, where $\small{uc\approx57.382\%}$¹⁵⁷), i.e., the expansion is limited to 69.84 – 143.19% or 67.7 – 150.0%, respectively (see this article for the contradiction with $\small{uc=57.4\%}$ given in the Canadian guidance).
3. It has to be demonstrated that the high variability of the reference is not caused by ‘outliers’.
  It should be noted that large deviations between geometric mean ratios arise as a natural, direct consequence of the high variability. Since extreme values are common for HVD(P)s, assessment of ‘outliers’ is not required by Brazil’s ANVISA and Chile’s ANAMED.

In all methods a point estimate-constraint is imposed. Even if a study would pass the scaled limits, the PE has to lie within 80.00 – 125.00% in order to pass. Whilst the PE-constraint is statistically not justified, it was implemented in all jurisdictions ‘for political reasons’.¹⁶⁴

“

There is no scientific basis or rationale for the point estimate recommendations

There is no belief that addition of the point estimate criteria will improve the safety of approved generic drugs

The point estimate recommendations are only “political” to give greater assurance to clinicians and patients who are not familiar (don’t understand) the statistics of highly variable drugs

Leslie Z. Benet (2006)¹⁶⁴

Compared to ABE, SABE leads to a substantial reduction in sample sizes (see this article). However, both RSABE and ABEL may result in an inflated type I error (the patient’s risk),¹⁰⁸ which was already described in 2009¹⁶² ¹⁶⁵ (before [sic] SABE was implemented) and is still an unresolved issue¹⁶⁶ ¹⁶⁷ (see also this article).

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RSABE

The FDA published SAS code^{154 161} but it is a mystery why a fixed-effects model for the partial replicate design and a mixed-effects model for a full replicate design was recommended. If you understand why, please let me know.

If $\small{s_\text{wR}<0.294}$, ABE has to be assessed by $\small{(7)}$ and $\small{\Delta=20\%}$ (90% CI entirely within 80.00 – 125.00%).

Over-specification: More parameters than could be uniquely estimated.

It must be mentioned that if the study was performed in a partial replicate design, the model is over-specified and the optimizer of any (‼) software might not converge (for details see this article).

If $\small{s_\text{wR}\geq0.294}$, RSABE should be applied. The regulatory constant is given by \[\theta_\text{s}=\frac{\log_{e}1.25}{s_0}\approx 0.8925742\ldots\small{\textsf{,}}\tag{14}\] where $\small{s_0}$ is the regulatory switching condition $\small{0.25}$. The point estimate $\small{PE}$ is given by $\small{\overline{Y}_\text{T}-\overline{Y}_\text{R}}$, where $\small{\overline{Y}_\text{T}}$ and $\small{\overline{Y}_\text{R}}$ are the means of $\small{\log_{e}}$-transformed PK-metrics obtained for the test and reference products, respectively. The standard error $\small{se}$ of the $\small{PE}$ is \[se=\sqrt{\frac{\widehat{s}}{{N_{s}}^{2}}\sum \frac{1}{n_i}}\small{\textsf{,}}\tag{15}\] where $\small{\widehat{s}}$ is the model’s residual mean squares error, $\small{N_\text{s}}$ are the number of sequences, and $\small{n_i}$ the number of subjects in sequence $\small{i}$. We start with the SABE model $\small{(13)}$ and work with $\small{\log_{e}\textsf{-}}$transformed values for convenience \[-\theta_\text{s}\leq\frac{\mu_\text{T}-\mu_\text{R}}{\sigma_\text{wR}}\leq\theta_\text{s}\tag{16}\] and use its squared and linearized form \[\left(\mu_\text{T}-\mu_\text{R}\right)^2-{\theta_{s}}^{2}\cdot{\sigma_{\text{wR}}}^{2}\leq0\small{\text{.}}\tag{17}\] Upon inspecting part of the SAS code in the FDA’s guidance…¹⁶¹

  pointest=exp(estimate);
  x=estimate**2-stderr**2;
  theta=((log(1.25))/0.25)**2;
  y=-theta*s2wr;

…we see that stderr**2, i.e., $\small{se^2}$ from $\small{(15)}$, is inserted in the left-hand side of $\small{(17)}$ – which is formulated in the true parameters – yielding for the estimates \[PE^2-se^2-{\theta_{s}}^{2}\cdot {s_{\text{wR}}}^{2}\leq0\small{\textsf{.}}\tag{18}\] This is not stated as such in the formulas of the guidance. We are aware of only one reference,¹⁶⁸ which is – regrettably – not in the public domain.

“The statistical approach we use is very similar to that proposed by Tothfalusi, Endrenyi, et al. 2001,¹⁶⁹ with a minor difference (use of an unbiased estimator for $\small{\left(\mu_\text{T}-\mu_\text{R}\right)^2})$.
Donald Schuirmann (2016)¹⁶⁸

Then \[\eqalign{ E_\text{m}&=PE^2-se^2\\ E_\text{s}&={\theta_{s}}^{2}-{s_{\text{wR}}}^{2} }\tag{19}\] are calculated, where $\small{E_\text{m}}$ and $\small{E_\text{s}}$ are the estimates of the true parameters ($\small{se^2}$ acts again as a bias correction). Since their distributions are known, their upper confidence limits $\small{C_\text{m}}$ and $\small{C_\text{s}}$ can be calculated by \[\eqalign{ C_\text{m}&=\left(\left|PE\right|+t_{1-\alpha,\,\nu}\cdot se\right)^2\\ C_\text{s}&=E_\text{s}\cdot \nu\big{/}\chi_{1-\alpha,\,\nu}^{2}\small{\textsf{,}} }\tag{20}\] where $\small{\nu}$ are the degrees of freedom given by $\small{\sum n-N_\text{s}}$. A modification¹⁷⁰ of Howe’s approximation¹⁷¹ is used in order to get the CI of a sum of random variables from the individual CIs. The squared lengths of the individual CIs are: \[\eqalign{ L_\text{m}&=\left(C_\text{m}-E_\text{m}\right)^2\\ L_\text{s}&=\left(C_\text{s}-E_\text{s}\right)^2\small{\textsf{.}} }\tag{21}\] Finally we calculate the 95% upper confidence bound: \[\small{\textsf{bound}}=E_\text{m}-E_\text{s}+\sqrt{\left(L_\text{m}-L_\text{m}\right)^2}\tag{22}\]

In order to pass RSABE:

$\small{\textsf{bound}}$ by $\small{(22)}$ has to be $\small{\leq0}$.
The PE has to lie within 80.00 – 125.00%.

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ABEL

Although the EMA’s concept paper stated¹⁵² that the statistical and computational methods will be given in the guideline, this was not the case.¹⁵³ SAS code and two example data sets were published later in a Q&A document.¹⁷² The evaluation has to be done with a simple ANOVA, i.e., assuming identical within-subject variances of the test and reference products. Methods to identify and handle outliers were not given.

If $\small{CV_\text{wR}\leq30\%}$, ABE has to be demonstrated by $\small{(7)}$ and $\small{\Delta=20\%}$ (90% CI entirely within 80.00 – 125.00%).

Otherwise, ABEL can be applied and the limits expanded to $\small{\left\{L,U\right\}=100\exp(\mp k\cdot s_\text{wR})}$, with the regulatory constant $\small{k=0.76}$. The scaling is capped at 50% for all agencies (maximum expansion 69.84 – 143.19%), except for Health Canada at ≈57.382% (67.7 – 150.0%).

invisible(library(PowerTOST))
CVwR      <- 100 * sort(c(seq(0.3, 0.6, 0.05), 0.57382))
EL        <- data.frame(CVwR   = CVwR,
                        EMA.uc = c(rep("no", 4), rep("yes", 4)),
                        EMA.L  = NA_real_, EMA.U = NA_real_,
                        HC.uc  = c(rep("no", 6), rep("yes", 2)),
                        HC.L   = NA_real_, HC.U  = NA_real_)
EMA       <- scABEL(CV = CVwR / 100, regulator = "EMA")
HC        <- scABEL(CV = CVwR / 100, regulator = "HC")
EL[, 1]   <- sprintf("%.3f%%", EL[, 1])
EL[, 3:4] <- sprintf("%.2f%%", 100 * EMA)
EL[, 6:7] <- sprintf("%.1f%%", 100 * HC)
names(EL)[2:7] <- c("capped", "L (EMA)", "U (EMA)",
                    "capped", "L (HC)", "U (HC)")
print(EL, row.names = FALSE)

#     CVwR capped L (EMA) U (EMA) capped L (HC) U (HC)
#  30.000%     no  80.00% 125.00%     no  80.0% 125.0%
#  35.000%     no  77.23% 129.48%     no  77.2% 129.5%
#  40.000%     no  74.62% 134.02%     no  74.6% 134.0%
#  45.000%     no  72.15% 138.59%     no  72.2% 138.6%
#  50.000%    yes  69.84% 143.19%     no  69.8% 143.2%
#  55.000%    yes  69.84% 143.19%     no  67.7% 147.8%
#  57.382%    yes  69.84% 143.19%    yes  66.7% 150.0%
#  60.000%    yes  69.84% 143.19%    yes  66.7% 150.0%

It has to be demonstrated that the high $\small{CV_\text{wR}}$ is not caused by outliers. If outliers are detected, they have to be excluded and $\small{CV_\text{wR}}$ as well as $\small{\left\{L,U\right\}}$ recalculated. However, the 90% CI has to be calculated with complete data.

In order to pass ABEL:

The 90% CI has to lie entirely within $\small{\left\{L,U\right\}}$.
The PE has to lie within 80.00 – 125.00%.

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NTIDs

Based on $\small{\Delta=10\%}$ for Narrow Therapeutic Index Drugs (EMA) and Critical Dose Drugs (Health Canada) the BE limits may need to be narrowed^{153 156 157} or scaled.¹⁶¹ ¹⁷³

For the EMA this means generally acceptance limits of 90.00 – 111.11% for $\small{AUC}$ only. Where $\small{C_\text{max}}$ is of particular importance for safety, efficacy or drug level monitoring, 90.00 – 111.11% should be applied as well. It is not possible to define a set of criteria to categorize drugs as NTIDs and it must be decided case by case based on clinical considerations whether an active substance is an NTID. However, according to all PSGs published so far, 90.00 – 111.11% is recommended only for $\small{AUC}$.
For Health Canada acceptance limits are 90.0 – 112.0% (see also this article) for $\small{AUC}$, whereas for $\small{C_\text{max}}$ the 90% CI has to be assessed for the conventional limits of 80.0 – 125.0%.

The FDA and China’s CDE require RSABE based on $\small{s_\text{wR}}$ (except dabigatran,¹⁵⁹ ¹⁷⁴ rivaroxaban,¹⁷⁵ ¹⁷⁶ and edoxaban,¹⁷⁷ which are considered ‘highly variable with a steep exposure-response relationship for both efficacy and safety’; therefore, RSABE must not be applied).
The study has to be performed in a two treatment two sequence four period full replicate design, thus allowing a comparison of $\small{s_\text{wT}}$ with $\small{s_\text{wR}}$.

With the regulatory switching condition $\small{s_0=0.10}$ we get the regulatory constant by \[\theta_\text{s}=\frac{\log_{e}1.11111}{s_0}\approx 1.053595\ldots\tag{23}\] The 95% upper confidence bound is determined with $\small{\theta_\text{s}}$ by $\small{(15)-(22)}$.
The upper CL for $\small{\sigma_\text{wT}/\sigma_\text{wR}}$ is calculated by \[\frac{s_\text{wT}/s_\text{wR}}{\sqrt{F_{{1-\alpha/2},\nu_1,\nu_2}}}\small{\textsf{,}}\tag{24}\] where $\small{s_\text{wT}}$ ist the estimate of $\small{\sigma_\text{wT}}$ with $\small{\nu_1}$ degrees of freedom, $\small{s_\text{wR}}$ ist the estimate of $\small{\sigma_\text{wR}}$ with $\small{\nu_2}$ degrees of freedom, and $\small{F}$ is the value of the F-distribution with $\small{\nu_1}$ (numerator) and $\small{\nu_2}$ (denominator) for $\small{\alpha=0.1}$.

In order to pass:

$\small{\textsf{bound}}$ by $\small{(22)}$ has to be $\small{\leq0}$.
The upper CL by $\small{(24)}$ has to be $\small{\leq 2.5}$.
ABE has to be demonstrated by $\small{(7)}$ and $\small{\Delta=20\%}$ (90% CI entirely within 80.00 – 125.00%).

The last condition is operationally equivalent to capping the ‘implied’ limits $\small{\left\{L,U\right\}}$ of RSABE at $\small{CV_\text{wR}\geq}$ $\small{\approx21.42\%}$. Otherwise, for any larger $\small{CV_\text{wR}}$ they would by wider than 80.00 – 125.00%. Of course, that is not what we want for an NTID. We can show that numerically.

fun <- function(x, Delta, sigma.0) { # x is CVwR
  theta.s   <- log(Delta) / sigma.0  # regulatory constant
  swR       <- sqrt(log(x^2 + 1))    # within subject standard deviation of R
  U         <- exp(theta.s * swR)    # upper ‘implied’ (scaled) limit
  objective <- U - 1.25              # target zero
  return(objective)
}
Delta   <- 1.11111 # approximate acc. to the guidance (not the exact 1/0.9)
sigma.0 <- 0.10    # regulatory switching condition
# numerically find the CVwR where U ~1.25
CVcap   <- 100 * uniroot(fun, interval = c(0, 0.3), tol = 1e-8,
                         Delta, sigma.0)$root
# check the ‘implied’ limits
CVwR    <- sort(c(CVcap / 100, seq(0.05, 0.3, 0.05)))
comp    <- data.frame(CVwR = CVwR, L.implied = NA_real_, U.implied = NA_real_,
                      L.capped = NA_real_, U.capped = NA_real_)
f       <- c(-1, +1)
for (i in seq_along(CVwR)) {
  comp[i, 2:5] <- sprintf("%.2f%%", 100 * exp(f * log(Delta) / sigma.0 *
                                              sqrt(log(CVwR[i]^2 + 1))))
  if (comp$CVwR[i] >= CVcap / 100) {
    comp[i, 4:5] <- sprintf("%.2f%%", 100 * exp(f * log(Delta) / sigma.0 *
                                                sqrt(log((CVcap / 100)^2 + 1))))
  }
}
comp$CVwR <- sprintf("%.2f%%", 100 * comp$CVwR)
txt       <- sprintf("The ‘implied’ limits in RSABE are capped at CVwR %.9g%%.\n", CVcap)
cat(txt); print(comp, row.names = FALSE)

# The ‘implied’ limits in RSABE are capped at CVwR 21.4189888%.
#    CVwR L.implied U.implied L.capped U.capped
#   5.00%    94.87%   105.41%   94.87%  105.41%
#  10.00%    90.02%   111.08%   90.02%  111.08%
#  15.00%    85.46%   117.02%   85.46%  117.02%
#  20.00%    81.17%   123.20%   81.17%  123.20%
#  21.42%    80.00%   125.00%   80.00%  125.00%
#  25.00%    77.15%   129.62%   80.00%  125.00%
#  30.00%    73.40%   136.25%   80.00%  125.00%

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BCS-based Biowaivers

Introduced by the FDA in 2000,¹⁷⁸ ¹⁴⁸ the EMA in 2010,¹⁵³ and the ICH in 2019¹⁷⁹ as an alternative for in vivo testing of IR products based on the Biopharmaceutic Classification System, where drugs are classified by their solubility and permeability.⁸⁸

Class I	Class II
High solubility	Low solubility
High permeability	High permeability
Class III	Class IV
High solubility	Low solubility
Low permeability	Low permeability

The idea behind waiving an in vivo study is based on the fact that such studies are not required for aqueous solutions (see above). Thus, if a drug product dissolves very rapidly, it can be expected to behave similarly to a solution.

A BCS-based biowaiver may be acceptable if the drug substance has been proven to exhibit high solubility and complete absorption (Class I) and either very rapid (> 85% within 15 min) or similarly rapid (85% within 30 min) in vitro dissolution characteristics of the test and reference product has been demonstrated considering specific requirements and excipients that might affect BA are qualitatively and quantitatively the same. In general, the use of the same excipients in similar amounts is preferred.

BCS-based biowaivers may also be acceptable if the drug substance has been proven to exhibit high solubility and limited absorption (Class III) and very rapid (> 85% within 15 min) in vitro dissolution of the test and reference product has been demonstrated considering specific requirements, excipients that might affect BA are qualitatively and quantitatively the same, and other excipients are qualitatively the same and quantitatively very similar.

The following conditions should be employed in the comparative dissolution studies to characterize the dissolution profile of the products:¹⁷⁹

Paddle of basket apparatus.
Volume of dissolution medium: ≤ 900 mL; preferrably the volume of the QC test.
Temperature of the dissolution medium: 37 ±1 ℃.
Agitation: 50 rpm (paddle apparatus), 100 rpm (basket apparatus).
≥ 12 units of the reference and test product.
Three buffers: pH 1.2, 4.5, and 6.8. Pharmacopoeial buffers should be employed. Additional investigation may be required at the pH of minimum solubility (if different from the buffers above).
Organic solvents are not acceptable and no surfactants should be added.
Samples should be filtered during collection, unless an in-situ detection method is used.
For gelatin capsules or tablets with gelatin coatings where cross-linking has been demonstrated, the use of enzymes may be acceptable, if appropriately justified.

When high variability or coning is observed in the paddle apparatus at 50 rpm for both reference and test products, the use of the basket apparatus at 100 rpm is recommended. Additionally, alternative methods (e.g., the use of sinkers or other appropriately justified approaches) may be considered to overcome issues such as coning, if scientifically substantiated.¹⁷⁹

The evaluation of the similarity factor $\small{f_2}$ is based on the following conditions:¹⁷⁹

A minimum of three time points (zero excluded).
The time points should be the same for the two products.
Mean of the individual values for every time point for each product.
Not more than one mean value of ≥ 85% dissolved for either of the products.
The coefficient of variation (CV) of mean values should not > 20% at early time points (≤ 10 min) and should not > 10% at other time points. When the CV is too high, the ƒ₂ calculation is considered inaccurate and a conclusion on similarity in dissolution cannot be made.

A risk assessment of potential bioinequivalence by application of a biowaiver has be provided, which has to be more strict for Class III than for Class I drugs.¹⁵³ Biowaivers for NTIDs are not possible.

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Open Issues

Alas, approaches are not harmonized yet^166
167 ¹⁸⁰ ¹⁸¹ ¹⁸² – after eight BioInternational conferences 1989 – 2008, six GBHI workshops 2015 – 2024, and more than five years of involvement of the ICH… At least there is an agreement to use a 90% CI.

Agency	Uncomplicated drug	HVD(P)	NTID
FDA ¹⁶¹	ABE (any metric) CI $\small{\subset}$ 80.00–125.00%	RSABE (any metric) $\small{\textsf{bound}\leq}$ 0 PE $\small{\subset}$ 80.00–125.00%	RSABE (any metric) $\small{\textsf{bound}\leq}$ 0 CI $\small{\subset}$ 80.00–125.00% upper CL of $\small{\sigma_\text{wT}/\sigma_\text{wR} \leq}$ 2.5
EMA ^{153 131}	ABE (any metric) CI $\small{\subset}$ 80.00–125.00%	ABEL ($\small{C_\text{max}}$, $\small{\textsf{p}AUC}$) $\small{uc}$ 50% PE $\small{\subset}$ 80.00–125.00%	ABE CI $\small{\subset}$ 90.00–111.11% PSGs: Only for $\small{AUC}$
WHO ¹⁵⁶ ¹⁸³	ABE (any metric) CI $\small{\subset}$ 80.00–125.00%	ABEL ($\small{C_\text{max}}$, $\small{AUC}$) $\small{uc}$ 50% PE $\small{\subset}$ 80.00–125.00%	ABE CI $\small{\subset}$ 90.00–111.11%
HC ¹⁵⁷	ABE $\small{AUC}$: CI $\small{\subset}$ 80.0–125.0% $\small{C_\text{max}}$: PE $\small{\subset}$ 80.0–125.0%	ABEL ($\small{AUC}$, $\small{uc}$ 57.382%) ABE ($\small{C_\text{max}}$: PE $\small{\subset}$ 80.0–125.0%)¹⁵⁷	ABE $\small{AUC}$: CI $\small{\subset}$ 90.0–112.0% $\small{C_\text{max}}$: CI $\small{\subset}$ 80.0–125.0%

This lack of harmonization leads to the paradox (though hypothetical) situation that the same study will pass in one jurisdiction but fail in another.^{108 166 167 180
181 182}

Only with a few exceptions (i.e., in Australia, Canada, New Zealand, Singapore, South Africa, Switzerland, Taiwan, the UK, and in countries following the WHO’s guidelines^{156 183}) under certain conditions, the local reference must be used in comparative BA studies. Accepting a foreign reference or – even better – a ‘Global Comparator’ would be desirable in order to reduce the number of studies. Under current legislation, this is not possible in most countries.¹⁸⁴

Still unresolved, outlook:

Control of the type I error in SABE?^{108 162 166}
Outliers in ABEL: Why and how?^{152 153 156 157} ¹⁸⁵
Innovators should perform a study in a replicate design in order to obtain estimates of the within-subject variability of PK metrics.^{105 107} This would allow to define fixed wider BE limits in PSGs and strictly control the type I error.^{108 167}
Method for NTIDs: Fixed narrower acceptance limits^{153 155 157} or reference-scaling?¹⁶¹ ¹⁸⁶ ¹⁸⁷
Comparison of ‘early exposure’¹⁸⁸ ¹⁸⁹ ¹⁹⁰ if clinically relevant (i.e., $\small{t_\text{max}}$ by a nonparametric method or first partial $\small{AUC}$)? See also this article.
Selection of cut-off times of partial $\small{AUC}\textsf{s}$ of multiphasic release products (i.e., based on PD – like the FDA or on PK – like the EMA)? If a PK/PD relationship is lacking, the selection of cut-off times is challenging at least.¹⁹¹
Assess only the global $\small{C_\text{max}}$ of multiphasic release products (like the FDA) instead of $\small{C_\text{max}}$ in each of the sections (like the EMA¹³⁰)?
Use a model with factors subject, period, treatment^{25 52 73 95} instead of the over-specified nested model given in all guidelines?
Remove¹⁹² assessment³ ¹⁹³ of the group-by-treatment interaction? In the final M13A guideline the ICH states that ‘potential for heterogeneity of treatment effect across groups’ should be evaluated and its impact on the study data discussed, ‘e.g., by investigation of group × treatment interaction in a supportive analysis and calculation of descriptive statistics by group.’¹⁹⁴ The ‘problem’ is elaborated in another article.
Is the questionable¹⁹⁵ ¹⁹⁶ ¹⁹⁷ recommendation for the inclusion of male and female subjects^{156 160} driven by ‘gender politics’ rather than science? ‘Inclusion of male and female subjects […] should be considered’.¹⁹⁴ THX a lot! However, in the October 2024 update of their PSGs (see below) the FDA ignored the consensus of ICH M13A and still recommends inclusion of both sexes.
Use $\small{C_\text{max}/AUC}$ as an alternative surrogate for the rate of absorption?^{38 39 40}
Decrease the variability of the extent of absorption of HVDs by using $\small{AUC/\widehat{t}_{1/2}}$ or $\small{AUC/\widehat{\lambda}_z}$?¹⁹⁸ ¹⁹⁹ ²⁰⁰ ²⁰¹ See also this article.
Use $\small{\widehat{C}_{\text{t}_\text{last}}}$¹⁸⁵ for the extrapolation of $\small{AUC}$; see also this article.
The requirement that $\small{AUC_{0-\text{t}}}$ of IR products has to cover ≥ 80% of $\small{AUC_{0-\infty}}$.^{153 156 193
194} It appeared out of blue skies already in the APV guideline of 1987⁷⁰ without any justification. Thoughtless copy & paste ever since? It is questionable because at $\small{2-4\,\times}$ $\small{t_\text{max}}$ absorption is practically complete.²⁰² ²⁰³
While for all IR products $\small{AUC_{0-72\text{h}}}$ (instead of $\small{AUC_{0-\text{t}}}$) is acceptable for the EMA and the WHO,^{153 156} the FDA, Health Canada, and the ICH accept that only for drugs with a half-life of > 24 hours.^{157 160 192
194} Why?
The elastic clause »appropriate sample size«^{154 194} was borrowed from ICH M9.²⁰⁴ However, M9 was intended for a statistical audience knowing what that means. The 2001 Note for Guidance of the EMEA,⁹⁷ as well as the ones of the WHO¹⁵⁶ and Health Canada¹⁸⁵ are more specific.
Testing of sequence, period and formulation effects with an »explanation« for significant effects¹⁸⁵ is ludicrous. Period effects mean out in crossover studies; see this article about sequence effects as well as that one about treatment effects.
‘Any concentration reported as below the lower limit of quantification (LLOQ) should be treated as zero in PK parameter calculations.’¹⁹⁴ I beg your pardon?
»After a dose we know only one thing for sure: The concentration is not zero.«²⁰⁵
Assessment whether steady-state has been achieved by comparing at least three pre-dose concentrations.^{130 194} See this article why it is problematic.
Should the requirement of multiple dose studies of extended release products^{1309 156} be abandoned?¹⁶¹ ²⁰⁶ ²⁰⁷
Assess $\small{C_\text{ss,min}}$ in multiple dose studies not only for originator’s products but also for generics?¹³¹
Adaptive sequential Two-Stage Designs (TSDs).²⁰⁸ ²⁰⁹ ²¹⁰ ²¹¹ Only exact²¹² ²¹³ or simulation-based methods²¹⁴ ²¹⁵ ²¹⁶ ²¹⁷ ²¹⁸ as well?
Limit the potency-correction^{153 156 194} not only to cases where measured contents differ by more than 5%?⁷⁵ The Canadian requirement¹⁵⁷ of demonstrating BE on both potency‐corrected and uncorrected data is peculiar.
Use Bayesian methods^{53 54 [55(#fn55) 56 57} ²¹⁹ ²²⁰ instead of the CI inclusion approach? Not surprisingly authors of the FDA argued against.²²¹

It is beyond me why the EMA’s guideline¹⁵³ (based on the European legislation²²²) refers – apart from salts and esters – to different ethers.

Different salts dissociate to the same base and different anions.
Different esters hydrolize to the same base and different alcohols.
Cleavage of ethers is simply impossible in physiological conditions.
This statement demonstrates a lack of understanding basic organic chemistry. Different ethers are different active moieties!

No other jurisdiction contains such a ludicrous statement.

“Those people who think they know everything
are a great annoyance to those of us who do.
attributed to Isaac Asimov

</nitpick>

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M13

At the Assembly of the International Council for Harmonisation (Singapore, 19/20 November 2019) the new topic ‘Bioequivalence for Immediate-Release Solid Oral Dosage Forms’ was proposed by the FDA. The Assembly approved the outline of a Concept Paper and agreed to establish a Working Group without delay to initiate work on finalizing the Concept Paper and Business Plan for the M13 topic.²²³ The Working Group was established in February 2020 and the Concept Paper endorsed in July 2020.²²⁴

The following stakeholders (31 members) are represented in the M13A Expert Working Group: ANVISA (1), EC (2), EFPIA (2), FDA (3), Global Self-Care Federation (1), Health Canada (1), HSA (1), IFPMA (1), IGBA (2), JFDA (1), JPMA (2), MFDS (1), MHRA (1), NAFDAC (1), NMPA (2), PhRMA (2), PMDA (2), SAHPRA (1), Swissmedic (1), TFDA (1), TGA (1), WHO (1).

Fig. 7 The ICH’s guidelines are developed in a step-wise procedure.

“Step 4: Adoption of an ICH Harmonised Guideline
Step 4 is reached when the Assembly agrees that there is sufficient consensus on the draft Guideline.
The Step 4 Final Document is adopted by the ICH Regulatory Members of the ICH Assembly as an ICH Harmonised Guideline at Step 4 of the ICH process.

“Step 5: Implementation
Having reached Step 4, the harmonised Guideline moves immediately to the final step of the process that is the regulatory implementation. This step is carried out according to the same national/regional procedures that apply to other regional regulatory guidelines and requirements, in the ICH regions.
ICH (2023)²²⁵ (my emphasis)

Although a guideline is in implementation, it is not necessarily implemented in all regions. There is no deadline and no obligation for an agency to implement it at all.

<#MeToo>

It is beyond me why some guidelines are not implemented for years, even by founding members of the ICH. As an example – the pretty straightforward – ‘Bioanalytical Method Validation and Study Sample Analysis’ guideline¹²⁸ of May 2022 is still not implemented in Japan. M10 is not rocket science…

</#MeToo>

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M13A

The draft guideline was published in December 2022.³ The ICH’s Assembly (Fukuoka, 4/5 June 2024) approved the final version, which was published together with Q&As in July 2024;¹⁹⁴ ²²⁶ see also a brief summary.²²⁷ A comparison of some PK- and statistics-ralated issues of the draft and final versions is given (together with my personal views) in the BEBA-Forum.²²⁸

Given the considerable degree of conformity already achieved by local guidelines in the mid-2010s, the development of a harmonized guideline should have been a relatively straightforward process. Furthermore, in the workshops of the Global Bioequivalence Harmonisation Initiative (Amsterdam 2015, 2018, 2022; Rockville 2016, 2024; Bethesda 2019) controversial issues and potential solutions were discussed in great detail. However, development of the guideline still took three and a half years.

The biggest difference to all previous guidelines is the definition of so-called “high-risk products”. For these products there is an increased likelihood that the in vivo performance will be affected differently by varying GI conditions in fasting and fed state. A rationale should be provided for the selection of the type of study(ies) – fasting, fed, or both – and meal type, e.g., fat and calorie content, based on the understanding of the test and comparator products.

High-risk products
Studies should be conducted under both fasting and fed conditions, irrespective of the drug product labelling with regard to food intake, if safety permits.
Non high-risk products
- If a drug product is labelled to be taken only under fasting conditions or can be taken under fasting or fed conditions, i.e., without regard to food, a single study conducted under fasting condition is recommended.
- If a drug product is labelled to be taken only with food due to PK reasons, i.e., enhancing absorption or reducing variability, a single study conducted under fed condition is recommended.
- If a drug product is labelled to be taken only with food due to tolerability reasons, e.g., stomach irritation or other non-PK reasons, a single study conducted under either fasting or fed condition is acceptable.

Due to the tiered approach of M13, different implementation dates in regions, whether Product-Specific Guidances (PSGs) exist and will be updated, contradictions with current guidelines will exist.²²⁹

Implementation Status

Implemented

Like all ICH guidelines, it was automatically implemented²³⁰ in Switzerland after reaching Step 4.
The guideline was adopted by the EMA in July 2024²³¹ ²³² and will be effective with 25 January 2025 – superseding applicable parts of the 2010 guideline¹⁵³ related to design considerations and data analysis for non-replicate studies.
In August 2024 the FDA published a list of 826 PSGs which were planned to be updated in October 2024.²³³ The FDA published its guidance and the Q&A in October 2024.²³⁴ ²³⁵ Contrary to the previous announcement, only 814 PSGs have been updated with 31 October 2023. Unfortunately the full list²³⁶ does not give links to the respective PSG.
For ≈14% of products the recommendation of performing studies in fasting condition was removed. For ≈86% of products the recommendation of performing studies in fed condition was removed. For the latter there are two products (pimavanserin tartrate, tablet and capsule), where a BCS-based biowaiver was added as an option.
According to the FDA’s overarching guidance¹⁶¹ AUC_0–inf is a mandatory PK metric, while it is not recommended in M13A.²³⁴ Which one prevails?

Implementation process

Canada (HPFB) and China (NMPA).

Not implemented

Argentina (ANMAT), Australia (TGA), Brazil (ANVISA), Egypt (EDA), Japan (PMDA), Jordan (JFDA), Korea (MFDS), Mexico (COFEPRIS), Nigeria (NAFDAC), Saudi Arabia (SFDA), Singapore (HSA), South Africa (SAHPRA), Taiwan (TFDA), Turkey (TITCK), and the U.K. (MHRA).

top of section ↩︎ previous section ↩︎

M13B/C

At the ICH’s Assembly (Incheon, 15/16 November 2022) the EWG reported that preliminary work on M13B ‘Additional Strength Biowaiver’ has commenced and Steps 1 and 2a/b were targeted for June 2023.²³⁷ Due to delays in the finalization of M13A, the draft of M13B was expected to be published for public consultation (Step 2b) in July 2024.²³⁸ However, according to the EWG’s Workplan of August 2024 public consulation of M13B is expected to start in February 2025, together with inital work on M13C, i.e., BE assessment for HVDs, NTIDs, and advanced data analysis considerations (reference-scaling, Two-Stage adaptive designs).²³⁹

top of section ↩︎ previous section ↩︎

TODO… (not covered in the article yet)

Line-extensions and dose-proportionality biowaivers^{153 156 161} ²⁴⁰ ²⁴¹ ²⁴²
Dissolution similarity²⁴³ ²⁴⁴ ²⁴⁵ ²⁴⁶ ²⁴⁷ ²⁴⁸ ²⁴⁹
In Vitro-In Vivo Correlation^130
153 ²⁵⁰ ²⁵¹ ²⁵² ²⁵³ ²⁵⁴ ²⁵⁵ ²⁵⁶ ²⁵⁷
Fixed Dose Combination (FDC) products¹⁹⁴ ²⁵⁸ ²⁵⁹ ²⁶⁰ ²⁶¹
BCS updates²⁶² ²⁶³ ²⁶⁴ ²⁶⁵ ²⁶⁶
Transdermal Drug Delivery Systems^{130 161}
Orally Inhaled Products²⁶⁷ ²⁶⁸ ²⁶⁹
BE-drift²⁷⁰ ²⁷¹ ²⁷² ²⁷³
Locally applied & locally acting (LALA) products²⁷⁴ ²⁷⁵ ²⁷⁶ ²⁷⁷ ²⁷⁸ ²⁷⁹ ²⁸⁰
Global Comparator^{117 184} ²⁸¹ ²⁸² ²⁸³ ²⁸⁴ ²⁸⁵ ²⁸⁶
Adjusted Indirect Comparisons²⁸⁷ ²⁸⁸ ²⁸⁹ ²⁹⁰ ²⁹¹ ²⁹² ²⁹³
Two-Stage^{208 209 210 211 212
213 214 215 216 217
218} ²⁹⁴ and Group-Sequential Designs²¹³ ²⁹⁵
Orphan Medicinal Products²⁹⁶ ²⁹⁷
Biosimilars²⁹⁸ ²⁹⁹ ³⁰⁰ ³⁰¹ ³⁰² ³⁰³ ³⁰⁴
Drug-Drug Interactions^{81 199} ³⁰⁵ ³⁰⁶ ³⁰⁷ ³⁰⁸ ³⁰⁹
Software validation³¹⁰ ³¹¹ ³¹² ³¹³ ³¹⁴
Data integrity/manipulation³¹⁴ ³¹⁵ ³¹⁶ ³¹⁷ ³¹⁸ ³¹⁹ ³²⁰

See also a – somewhat outdated – collection of guidelines, my presentations, and further readings.^{144 145
191} ³²¹ ³²² ³²³ ³²⁴ ³²⁵ ³²⁶ ³²⁷ ³²⁸ ³²⁹ ³³⁰ ³³¹ ³³² ³³³ ³³⁴ ³³⁵
To whom it may concern.³³⁶ ³³⁷ ³³⁸

The textbooks dealing mainly with statistics (marked with ★) are rather tough cookies and not recommended for beginners.

Acknowledgments

I would like to thank Henning Blume and José A. Guimarães Morais for sharing memories about the BioInternational conferences and the early period of bioequivalence. Special thanks go to my co-organizers of the BioBridges workshops: Jean-Michel Cardot, Vít Perlík, and Ondřej Slanař . I’m indebted to Paulo Paixão for encouraging me to pursue my PhD at the Faculty of Pharmacy, Universidade de Lisboa. I’m also grateful to Susana Almeida for insights into the generic industry.

Dedicated to my friend Dirk Barends (1945 – 2012), who initiated the FIP’s biowaiver monograph series. I miss his wit and laughter. In gratitude to László Endrényi (1933 – 2020), whose work on pharmacokinetic metrics and the bioequivalence of Highly Variable Drugs inspired many scientists in the field.

Licenses

Helmut Schütz 2025
, PowerTOST, Rüdiger Appel’s station-clock, and pandocGPL 3.0, klippy MIT.
1^st version April 9, 2024. Rendered January 11, 2025 10:44 CET by rmarkdown via pandoc in 0.14 seconds.

Abbreviations

Abbreviation	Meaning
AAPS	American Association of Pharmaceutical Scientists
ABE	Average Bioequivalence
ABEL	Average Bioequivalence with Expanding Limits
ANAMED	Agencia Nacional de Medicamentos (competent authority of Chile)
ANDA	Abbreviated New Drug Application (generics; FDA term)
ANMAT	Administración Nacional de Medicamentos, Alimentos y Tecnología Médica (competent authority of Argentina)
ANOVA	Analyis of Variance
ANVISA	Agência Nacional de Vigilância Sanitária (competent authority of Brazil)
AOAC	(U.S.) Association of Official Analytical Chemists
APhA	(U.S.) Academy of Pharmaceutical Sciences
API	Active Pharmaceutical Ingredient
APV	Arbeitsgemeinschaft für Pharmazeutische Verfahrenstechnik (International Association for Pharmaceutical Technology)
$\small{AUC}$	Area Under the (concentration-time) Curve
$\small{AUC_{0-\text{t}}}$	$\small{AUC}$ from the time of administration to the time of the last measurable concentration
$\small{AUC_{0-72\text{h}}}$	$\small{AUC}$ from the time of administration to 72 hours (IR products)
$\small{AUC_{0-\infty}}$	$\small{AUC}$ from the time of administration extrapolated to infinite time
BA	Bioavailability
BCS	Biopharmaceutic Classification System
BE	Bioequivalence
BfARM	Bundesinstitut for Arzneimittel und Medizinprodukte (competent authority of Germany)
$\small{\textsf{bound}}$	95% upper confidence bound in RSABE
$\small{C}$	Concentration
CDE	Center for Drug Evaluation (China)
CDER	Center for Drug Evaluation and Research (FDA)
CFR	Code of Federal Regulations (U.S.)
cGMP	curent Good Manufacturing Practices
CHMP	Committee for Medicinal Products for Human Use (of the EMA)
CI	Confidence Interval
$\small{CL}$	Clearance
CL	Confidence Limit
CL_lower, CL_upper	Lower and upper CL
$\small{C_\text{max}}$	Maximum concentration
COFEPRIS	Comisión Federal para la Protección contra Riesgos Sanitarios (competent authority of Mexico)
CPMP	Committee for Proprietary Medicinal Products (of the EMEA)
CRO	Contract Research Organization
$\small{C_\text{ss,min}}$	Minimum concentration in steady-state within the dosing interval $\small{\tau}$
$\small{C_{\text{ss}\,,\tau}}$	Concentration in steady-state at the end of the dosing interval $\small{\tau}$
$\small{C_{\text{t}_\text{last}}}$	Last measured concentration
$\small{\widehat{C}_{\text{t}_\text{last}}}$	Estimated concentration at $\small{t_\text{last}}$
CVM	Center for Veterinary Medicine (FDA)
$\small{CV_\text{intra}}$	Within-subject Coefficient of Variation in a crossover design
$\small{CV_\text{wR},CV_\text{wT}}$	Observed within-subject Coefficient of Variation of the Reference and Test product
$\small{D}$	Dose
EC	European Community
EEA	European Economic Area (EU + Liechtenstein, Iceland, Norway)
EFPIA	European Federation of Pharmaceutical Industries and Associations
EMA	European Medicines Agency
$\small{E_\text{max}}$	Maximum effect
EMEA	European Agency for the Evaluation of Medicinal Products (rebranded to EMA in Dec. 2009)
EOB	Electronic Orange Book (FDA)
EU	European Union
EUFEPS	European Federation for Pharmaceutical Sciences
EWG	Expert Working Group (ICH)
$\small{f}$	Fraction absorbed
$\small{f_2}$	Similarity factor
FDA	(U.S.) Food and Drug Administration
FDC	Fixed Dose Combination (product)
FIP	Federation International Pharmaceutique (International Pharmaceutical Federation)
GBHI	Global Bioequivalence Harmonisation Initiative
GI	Gastrointestinal
GLP	Good Laboratory Practice
GSD	Group-Sequential Design
$\small{H_0}$	Null hypothesis
$\small{H_1}$	Alternative hypothesis (also $\small{H_\text{a}}$)
HPFB	Health Products and Food Branch (competent authority of Canada)
HSA	Health Sciences Authority (competent authority of Singapore)
HVD(P)	Highly Variable Drug (Product)
IBE	Individual Bioequivalence
ICH	International Council for Harmonisation of Technical Requirements for Pharmaceuticals for Human Use
IFPMA	International Federation of Pharmaceutical Manufacturers and Associations
IGBA	International Generic and Biosimilar Medicines Association
IR	Immediate Release (product)
ISR	Incurred sample reanalysis
IV	Intravenous
JFDA	Jordan Food and Drug Administration
JPMA	Japan Pharmaceutical Manufacturers Association
$\small{k}$	Regulatory constant in ABEL: 0.76
$\small{k\,_\text{a}}$	Absorption rate constant (also $\small{k_\text{01}}$)
$\small{k\,_\text{el}}$	Elimination rate constant (also $\small{k_\text{10}}$)
$\small{L}$	Lower expanded limit in ABEL
LALA	Locally applied, locally acting (product)
LLOQ	Lower limit of quantification
$\small{\log_{e}}$	Natural logarithm (base e ≈ 2.71828…)
$\small{\log_{10}}$	Decadic logarithm (base 10)
$\small{L,U}$	Lower and upper expanded limits in ABEL
M	Multidisciplinary guideline of the ICH
MA	Market Authorisation
MEB	Medicines Evaluation Board (competent authority of The Netherlands)
MFDS	Ministry of Food and Drug Safety (competent authority of the Republic of Korea)
MHRA	Medicines and Healthcare products Regulatory Agency (competent authority of the U.K.)
MR	Modified release (product)
$\small{n}$	Sample size
$\small{n_1,n_2}$	Number of subjects in sequences 1 and 2 of a 2×2×2 crossover design
NAFDAC	National Agency for Food and Drug Administration and Control (competent authority of Nigeria)
NMPA	National Medical Products Administration (competent authority of China)
NDA	New Drug Application (originators; FDA term)
NTID	Narrow Therapeutic Index Drug (Canada: Critical Dose Drug)
OGD	Office of Generic Drugs (FDA)
OGDP	Office of Generic Drug Policy (FDA)
$\small{p(x)}$	Probability of $\small{x}$
$\small{\text{p}AUC}$	Partial $\small{AUC}$
PBE	Population Bioequivalence
PD	Pharmacodynamics
PE	Point Estimate of $\small{\mu_\text{T}/\mu_\text{R}}$
PhRMA	Pharmaceutical Research and Manufacturers of America
PMDA	Pharmaceuticals and Medical Devices Agency (competent authority of Japan)
PQT	Prequalification Team (WHO)
PK	Pharmacokinetics
PKWP	Pharmacokinetics Working Party (of the EMA’s CHMP)
PQT	Prequalification Team (WHO)
PSG	Product-Specific Guidance
Q&A	Question and Answer
$\small{\text{R}}$	Reference product
RLD	Reference Listed Drug (FDA term)
RMP	Reference Medicinal Product (EU term)
RSABE	Reference-Scaled Average Bioequivalence
$\small{s}$	Sample standard deviation
$\small{s_0}$	Switching condition in RSABE: for HVD(P)s 0.25 and for NTIDs 0.1
$\small{s^2}$	Sample variance
SABE	Scaled Average Bioequivalence
SAHPRA	South African Health Products Regulatory Authority
SUPAC	Scale-Up and Postapproval Changes (FDA)
$\small{s_\text{wR},s_\text{wT}}$	Observed within-subject standard deviation of the Reference and Test product
$\small{s_{\text{wR}}^{2},s_{\text{wT}}^{2}}$	Observed within-subject variance of the Reference and Test product
$\small{t}$	Time
$\small{\text{T}}$	Test product
TFDA	Taiwan Food and Drug Administration
$\small{\text{tc}}$	Cut-off time (multiphasic release products)
TE	Therapeutic Equivalence
TGA	Therapeutic Goods Administration (competent authority of Australia)
TIE	Type I Error
TITCK	Türkiye İlaç ve Tıbbi Cihaz Kurumu (competent authority of Turkey)
$\small{t_\text{last}}$	Time of the last measured concentration $\small{C_{\text{t}_\text{last}}}$
$\small{t_\text{max}}$	Time of $\small{C_\text{max}}$
TOST	Two One-Sided Tests
TSD	Two-Stage Design
$\small{U}$	Upper expanded limit in ABEL
$\small{uc}$	Upper cap of expansion in ABEL
URL	Uniform Resource Locator
$\small{V}$	Apparent volume of distribution
WHO	World Health Organization
$\small{\bar{x}_\text{T},\bar{x}_\text{R}}$	Arithmetic means of $\small{\text{T}}$ and $\small{\text{R}}$
$\small{\alpha}$	Nominal level of the test, probability of Type I Error (patient’s risk)
$\small{\beta}$	Probability of Type II Error (producer’s risk), where $\small{\beta=1-\pi}$
$\small{\Delta}$	Clinically relevant difference
$\small{\theta_\text{s}}$	Regulatory constant in RSABE: for HVD(P)s 0.8925742… and for NTIDs 1.053595…
$\small{\theta_0}$	True (in sample size estimation assumed) $\small{\text{T}/\text{R}}$-ratio
$\small{\theta_1,\theta_2}$	Fixed lower and upper limits of the BE acceptance range
$\small{\theta_{\text{s}_1},\theta_{\text{s}_2}}$	Scaled lower and upper limits of the BE acceptance range
$\small{\widehat{\lambda}_\text{z}}$	Apparent terminal rate constant (estimated)
$\small{\mu_\text{T}/\mu_\text{R}}$	True $\small{\text{T}/\text{R}}$-ratio
$\small{\nu}$	Degrees of freedom
$\small{\pi}$	Prospective (a priori) power, where $\small{\pi=1-\beta}$
$\small{\widehat{\pi}}$	Estimated (a posteriori, post hoc, retrospective) power
$\small{\sigma}$	Population standard deviation
$\small{\sigma_\text{wR},\sigma_\text{wT}}$	True within-subject standard deviation of the Reference and Test product
$\small{\tau}$	Dosing interval
2×2×2	Two treatment two sequence two period crossover design

Footnotes and References

I tried to give online-resources as far as possible. Others were published before the Internet was developed. I have them on yellowed or even faint thermal paper of (yes!) FAX-machines. Some books are out of print; perhaps you can get them used. No, I will not sell any of them.
I checked all URLs in December 2024. Contrary to us mere mortals who have to maintain a version control of documents, agencies don’t care. They change the structure of their websites (worst are the ones of the ANVISA and the WHO), don’t establish automatic redirects, rename or even delete files…
In 2009 the FDA restructured its website with only a few redirects. Which makes things worse still, the ‘new’ pages have even disappeared in the meantime. I’m a reviewer at the Wikipedia. It was a Sisyphean task to fix the more than 2,000 links acting as references in articles. Heck, we are doing that in our free time – THX,

!
Quod licet Iovi, non licet bovi
If you discover an error, please drop me a note at [email protected].

Vitti TG, Banes D, Byers TE. Bioavailability of Digoxin. N Engl J Med. 1971; 285(25): 1433–4. doi:10.1056/NEJM197112162852512.↩︎
DeSante KA, DiSanto AR, Chodos DJ, Stoll RG. Antibiotic Batch Certification and Bioequivalence. JAMA. 1975; 232(13): 1349–51. doi:10.1001/jama.1975.03250130033016.↩︎
ICH. Bioequivalence for Immediate-Release Solid Oral Dosage Forms. M13A. Draft version. 20 December 2022. Online.↩︎
Hall DG, In: Hearing Before the Subcommittee on Monopolies Select Committee on Small Business. U.S. Senate, Government Printing Office, Washington D.C. 1967: 258–81.↩︎
Tyrer JH, Eadie MJ, Sutherland JM, Hooper WD. Outbreak of anticonvulsant intoxication in an Australian city. Br Med J. 1970; 4: 271–3. doi:10.1136/bmj.4.5730.271. Open Access.↩︎
Bochner F, Hooper WD, Tyrer JH, Eadie MJ. Factors involved in an outbreak of phenytoin intoxications. J Neurol Sci. 1972; 16(4): 481–7. doi:10.1016/0022-510x(72)90053-6.↩︎
Lund L. Clinical significance of generic inequivalence of three different pharmaceutical preparations of phenytoin. Eur J Clin Pharmacol. 1974; 7: 119–24. doi:10.1007/bf00561325.↩︎
Lindenbaum J, Mellow MH, Blackstone MO, Butler VP. Variations in biological activity of digoxin from four preparations. N Engl J Med. 1971; 285(24): 1344–7. doi:10.1056/NEJM197112092852403.↩︎
Wagner JG, Christensen M, Sakmar E, Blair D, Yates JD, Willis PW 3^rd, Sedman AJ, Stoll RG. Equivalence lack in digoxin plasma levels. JAMA, 1973; 224(2): 199–204. PMID 4739492.↩︎
Lindenbaum J, Preibisz JJ, Butler VP Jr., Saha JR. Variation in digoxin bioavailabity: a continuing problem. J Chron Dis. 1973; 16: 749–54. doi:10.1056/nejm197112092852403.↩︎
Levy G, Gibaldi M. Bioavailability of Drugs. Focus on Digoxin. Circulation. XLIX(3); 1974: 391–4. doi:10.1161/01.CIR.49.3.391. Open Access.↩︎
Jounela AJ, Pentikäinen PJ, Sothmann. Effect of particle size on the bioavalability of digoxin. Eur J Clin Pharmacol. 1975; 8(5): 365–70. doi:10.1007/BF00562664.↩︎
Richton-Hewett S, Foster E, Apstein CS. Medical and Economic Consequences of a Blinded Oral Anticoagulant Brand Change at a Municipal Hospital. Arch Intern Med. 1988; 148(4): 806–8. doi:10.1001/archinte.1988.00380040046010.↩︎
Weinberger M, Hendeles L, Bighley L, Speer J. The Relation of Product Formulation to Absorption of Oral Theophylline. N Engl J Med. 1978; 299(16): 852–7. doi:10.1056/nejm197810192991603.↩︎
Bielmann B, Levac TH, Langlois Y, L Tetreault L. Bioavailability of primidone in epileptic patients. Int J Clin Pharmacol. 1974; 9(2): 132–7. PMID 4208031↩︎
Skelly JP, Knapp G. Biologic availability of digoxin tablets. JAMA. 1973; 224(2): 243. doi:10.1001/jama.1973.03220150051015.↩︎
Skelly JP. A History of Biopharmaceutics in the Food and Drug Administration 1968–1993. AAPS J. 2010; 12(1): 44–50. doi:10.1208/s12248-009-9154-8. Free Full Text.↩︎
APhA. Guidelines for Biopharmaceutic Studies in Man. Washington D.C. February 1972.↩︎
Skelly JP. Bioavailability and Bioequivalence. J Clin Pharmacol. 1976; 16(10/2): 539–45. doi:10.1177/009127007601601013.↩︎
Gardener S (Acting Commissioner of Food and Drugs). CFR, Title 21, Vol. 5, Chapter I, Part 320. Bioavailability and Bioequivalence Requirements. Procedures for Determining the In Vivo Bioavailability of Drug Products. December 30, 1976. Effective July 7, 1977, In: FR, Vol. 42, No. 5. January 7, 1977. Online.↩︎
Schuirmann DJ. A comparison of the Two One-Sided Tests Procedure and the Power Approach for Assessing the Equivalence of Average Bioavailability. J Pharmacokin Biopharm. 1987; 15(6): 657–80. doi:10.1007/BF01068419.↩︎
Metzler CM. Bioavailability – A Problem in Equivalence. Biometrics. 1974; 30(2): 309–17. PMID 4833140.↩︎
Westlake WJ. Symmetrical Confidence Intervals for Bioequivalence Trials. Biometrics. 1976; 32(4): 741–4. PMID 1009222.↩︎
Mantel N. Do We Want Confidence Intervals Symmetrical About the Null Value? Biometrics. 1977; 33: 759–60. [Letter to the Editor]↩︎
Westlake WJ. Design and Evaluation of Bioequivalence Studies in Man. In: Blanchard J, Sawchuk RJ, Brodie BB, editors. Principles and perspectives in Drug Bioavailability. Basel: Karger; 1979. ISBN 3-8055-2440-4. p. 192–210.↩︎
Fieller EC. Some Problems In Interval Estimation. J Royal Stat Soc B. 1954; 16(2): 175–85. doi:10.1111/j.2517-6161.1954.tb00159.x.↩︎
Locke CS. An Exact Confidence Interval from Untransformed Data for the Ratio of Two Formulation Means. J Pharmacokin Biopharm. 1984; 12(6): 649–55. doi:10.1007/bf01059558.↩︎
U.S. Department of Health and Human Services, FDA, Office of Medical Products and Tobacco, CDER, OGD, OGDP. Approved Drug Products with Therapeutic Equivalence Evaluations. Download.↩︎
U.S. Department of Health and Human Services, FDA, Office of Medical Products and Tobacco, CDER, OGD, OGDP. Approved Drug Products with Therapeutic Equivalence Evaluations. Cumulative Supplement. Download.↩︎
US Government Publishing Office. Drug Price Competition and Patent Term Restoration Act of 1984. Public Law 98-117. Sept. 24, 1984. Online.↩︎
In phase III we try to demonstrate that verum performs ‘better’ than placebo, i.e., one-sided tests for non-inferiority (effect) and non-superiority (adverse reactions). Such studies are already large: Approving statins and COVID-19 vaccines required ten thousands volunteers. Can you imagine how many it would need to detect a 20% difference between two treatments?↩︎
Benet LZ. Why Do Bioequivalence Studies in Healthy Volunteers? Presentation at: 1^st MENA Regulatory Conference on Bioequivalence, Biowaivers, Bioanalysis and Dissolution. Amman. 23 September 2013. Internet Archive.↩︎
Office of the Federal Register. Code of Federal Regulations, Title 21, Part 320, Subpart A, § 320.23(a)(1) Online.↩︎
This is an assumption, i.e., based on the labelled content instead of the measured potency.↩︎
Yet another assumption. Incorrect for highly variable drugs and, thus, inflates the confidence interval.↩︎
Tóthfalusi L, Endrényi L. Estimation of C_max and T_max in Populations After Single and Multiple Drug Administration. J Pharmacokin Pharmacodyn. 2003; 30(5): 363–85. doi:10.1023/b:jopa.0000008159.97748.09.↩︎
These formulas are only valid for a one-compartment model with zero order absorption and first order elimination. In all other models $\small{t_\text{max}}$ (and thus, $\small{C_\text{max}}$) cannot be analytically derived. In software numeric optimization is employed to locate the maximum of the function.↩︎
Endrényi L, Fritsch S, Yan W. C_max/AUC is a clearer measure than C_max for absorption rates in investigations of bioequivalence. Int J Clin Pharmacol Ther Toxicol. 1991; 29(10): 394–9. PMID 1748540.↩︎
Schall R, Luus HG. Comparison of absorption rates in bioequivalence studies of immediate release drug formulations. Int J Clin Pharmacol Ther Toxicol. 1992; 30(5): 153–9. PMID 1592542.↩︎
Endrényi L, Yan W. Variation of C_max and C_max/AUC in investigations of bioequivalence. Int J Clin Pharm Ther Toxicol. 1993; 31(4): 184–9. PMID 8500920.↩︎
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Cabana BE. Assessment of 75/75 Rule: FDA Viewpoint. Pharm Sci. 1983; 72(1): 98–9. doi:10.1002/jps.2600720127.↩︎
Haynes JD. FDA 75/75 Rule: A Response. Pharm Sci. 1983; 72(1): 99–100.↩︎
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Klingler D, Nitsche V, Schmidbauer H. Hydantoin-Intoxikation nach Austausch scheinbar gleichwertiger Diphenylhydantoin-Präparate. Wr Med Wschr. 1981; 131: 295–300. [German]↩︎
Glazko AJ, Chang T, Bouhema J, Dill WA, Goulet JR, Buchanan RA. Metabolic disposition of diphenylhydantoin in normal human subjects following intravenous administration. Clin Pharmacol Ther. 1969; 10(4): 498–504. doi:10.1002/cpt1969104498.↩︎
Bochner F, Hooper WD, Tyrer JH, Eadi MJ. Effect of dosage increments on blood phenytoin concentrations. J Neurol Neurosurg Psychiatr. 1972; 35(6): 873–6. doi:10.1136/jnnp.35.6.873.↩︎
Kirkwood TBL. Bioequivalence Testing—A Need to Rethink [reader reaction]. Biometrics. 1981, 37: 589—91. doi:10.2307/2530573.↩︎
Westlake WJ. Response to Bioequivalence Testing—A Need to Rethink [reader reaction response]. Biometrics. 1981, 37: 591–3.↩︎
Westlake WJ. Bioavailability and Bioequivalence of Pharmaceutical Formulations. In: Pearce KE, editor. Biopharmaceutical Statistics for Drug Development. New York: Marcel Dekker; 1988. ISBN 0-8247-7798-0. p. 329–53.↩︎
Rodda BE, Davis RL. Determining the probability of an important difference in bioavailability. Clin Pharmacol Ther. 1980; 28: 247–52. doi:10.1038/clpt.1980.157.↩︎
Mandallaz D, Mau J. Comparison of Different Methods for Decision-Making in Bioequivalence Assessment. Biometrics. 1981; 37: 213–22. PMID 6895040.↩︎
Fluehler H, Hirtz J, Moser HA. An Aid to Decision-Making in Bioequivalence Assessment. J Pharmacokin Biopharm. 1981; 9: 235–43. doi:10.1007/BF01068085.↩︎
Selwyn MR, Hall NR. On Bayesian Methods for Bioequivalence. Biometrics. 1984; 40: 1103–8. PMID 6398710.↩︎
Fluehler H, Grieve AP, Mandallaz D, Mau J, Moser HA. Bayesian Approach to Bioequivalence Assessment: An Example. J Pharm Sci. 1983; 72(10): 1178–81. doi:10.1002/jps.2600721018.↩︎
Anderson S, Hauck WW. A New Procedure for Testing Bioequivalence in Comparative Bioavailability and Other Clinical Trials. Commun Stat Ther Meth. 1983; 12(23): 2663–92. doi:10.1080/03610928308828634.↩︎
Steinijans VW, Diletti E. Statistical Analysis of Bioavailability Studies: Parametric and Nonparametric Confidence Intervals. Eur J Clin Pharmacol. 1983; 24: 127–36. doi:10.1007/BF00613939.↩︎
Steinijans VW, Diletti E. Generalization of Distribution-Free Confidence Intervals for Bioavailability Ratios. Eur J Clin Pharmacol. 1985; 28: 85–8. doi:10.1007/BF00635713.↩︎
Steinijans VW, Schulz H-U, Beier W, Radtke HW. Once daily theophylline: multiple-dose comparison of an encapsulated micro-osmotic system (Euphylong) with a tablet (Uniphyllin). Int J Clin Pharm Ther Toxicol. 1986; 24(8): 438–47. PMID 3759279.↩︎
Steinijans VW. Pharmacokinetic Characteristics of Controlled Release Products and Their Biostatistical Analysis. In: Gundert-Remy U, Möller H, editors. Oral Controlled Release Products – Therapeutic and Biopharmaceutic Assessment. Stuttgart: Wissenschaftliche Verlagsgesellschaft; 1988, ISBN 978-3804710429. p. 99–115.↩︎
Blume H, Siewert M, Steinijans V. Bioäquivalenz von per os applizierten Retard-Arzneimitteln; Konzeption der Studien und Entscheidung über Austauschbarkeit. Pharm Ind. 1989; 51: 1025–33. [German]↩︎
Wijnand HP, Timmer CJ. Mini-computer programs for bioequivalence testing of pharmaceutical drug formulations in two-way cross-over studies. Comput Programs Biomed. 1983; 17(1–2): 73–88. doi:10.1016/0010-468x(83)90027-2.↩︎
Where did it come from? Two stories:
Les Benet told that there was a poll at the FDA and – essentially based on gut feeling – the 20% saw the light of day.
I’ve heard another one, which I like more. Wilfred J. Westlake, one of the pioneers of BE was a statistician at SKF. During a coffee and cig break (everybody was smoking in the 1970s) he asked his fellows of the clinical pharmacology department »Which difference in blood concentrations do you consider relevant?« Yep, the 20% were born.↩︎
Rheinstein P. Report by the Bioequivalence Task Force on Recommendations from the Bioequivalence Hearing conducted by the Food and Drug Administration. September 29 – October 1986. January 1988.↩︎
APV. Richtlinie und Kommentar. Pharmazeutische Industrie. 1985; 47(6): 627–32. [German]↩︎
Arbeitsgemeinschaft Pharmazeutische Verfahrenstechnik (APV). International Symposium. Bioavailability / Bioequivalence, Pharmaceutical Equivalence and Therapeutic Equivalence. Würzburg. 9–11 February, 1987.↩︎
Junginger H. APV-Richtlinie – »Untersuchungen zur Bioverfügbarkeit, Bioäquivalenz« Pharm Ztg. 1987; 132: 1952–5. [German]↩︎
Junginger H. Studies on Bioavailability and Bioequivalence – APV Guideline. Drugs Made in Germany. 1987; 30: 161–6.↩︎
Nordic Council on Medicines. Bioavailabaility Studies in Man. Nordic Guidelines. Uppsala. 1^st edition October 1987. BEBAC Archive.↩︎
Blume H, Kübel-Thiel K, Reutter B, Siewert M, Stenzhorn G. Nifedipin: Monographie zur Prüfung der Bioverfügbarkeit / Bioäquivalenz von schnell-freisetzenden Zubereitungen (1). Pharm Ztg. 1988; 133(6): 398–93. [German]↩︎
TGA. Guidelines for Bioavailability and Bioequivalency Studies. Draft C06:6723c (29/11/88).↩︎
Blume H, Mutschler E. Bioäquivalenz – Qualitätsbewertung wirkstoffgleicher Fertigarzneimittel: Anleitung-Methoden-Materialien. Frankfurt/Main: Govi-Verlag; 1989. [German]↩︎
McGilveray IJ, Midha KK, Skelly JP, Dighe S, Doluiso JT, French IW, Karim A, Burford R. Consensus Report from “Bio International ’89”: Issues in the Evaluation of Bioavailability Data. J Pharm Sci. 1990; 79(10): 945–6. doi:10.1002/jps.2600791022.↩︎
Keene ON. The log transformation is special. Stat Med. 1995; 14(8): 811–9. doi:10.1002/sim.4780140810. Open Access.↩︎
Diletti E, Hauschke D, Steinijans VW. Sample size determination for bioequivalence assessment by means of confidence intervals. Int J Clin Pharm Ther Toxicol. 1991; 29(1): 1–8. PMID 2004861.↩︎
Diletti E, Hauschke D, Steinijans VW. Sample size determination: Extended tables for the multiplicative model and bioequivalence ranges of 0.9 to 1.11 and 0.7 to 1.43. Int J Clin Pharm Ther Toxicol. 1992; 30 (Suppl.1): S59–62. PMID 1601533.↩︎
Hauschke D, Steinijans VW, Diletti E. A distribution-free procedure for the statistical analysis of bioequivalence studies. Int J Clin Pharm Ther Toxicol. 1990; 28(2): 72–8.↩︎
Steinijans VW, Hauschke D. Update on the statistical analysis of bioequivalence studies. Int J Clin Pharm Ther Toxicol. 1990; 28(3): 105–10. PMID 2318545.↩︎
Steinijans VW, Hartmann M, Huber R, Radtke HW. Lack of pharmacokinetic interaction as an equivalence problem. Int J Clin Pharm Ther Toxicol. 1991; 29(8): 323–8. PMID 1835963.↩︎
Anderson S, Hauck WW. Consideration of individual bioequivalence. J Pharmacokinet Biopharm 1990; 18(3): 259–73. doi:10.1007/bf01062202.↩︎
Schall R, Luus HG. On population and individual bioequivalence. Stat Med 1993; 12(12): 1109–24. doi:10.1002/sim.4780121202.↩︎
Schall R. A unified view of individual, population, and average bioequivalence. In: Blume HH, Midha KK, editors. Bio-International 2. Bioavailability, Bioequivalence and Pharmacokinetic Studies. Stuttgart: medpharm; 1995. ISBN 3-88763-040-8. p. 91–106.↩︎
Chow S-C, Liu J-p. Design and Analysis of Bioavailability and Bioequivalence Studies. New York: Marcel Dekker; 1992. ISBN 0-8247-8682-3. ★↩︎
CPMP Working Party. Investigation of Bioavailability and Bioequivalence: Note for Guidance. III/54/89-EN, 8^th Draft. June 1990.↩︎
Commission of the EC. Investigation of Bioavailabilty and Bioequivalence. Brussels. December 1991. BEBAC Archive.↩︎
Amidon GL, Lennernäs H, Shah VV, Crison JR. A Theoretical Basis for a Biopharmaceutic Drug Classification: The Correlation of in Vitro Drug Product Dissolution and in Vivo Bioavailability. Pharm Res. 1995; 12(3): 413–20. doi:10.1023/a:1016212804288. Open Access.↩︎
Steinijans VW, Hauschke D. International Harmonization of Regulatory Bioequivalence Requirements. Clin Res Reg Aff. 1993; 10(4): 203–20.↩︎
FDA, CDER. Guidance for Industry. Statistical Procedures for Bioequivalence Studies using a Standard Two-Treatment Crossover Design. Rockville. Jul 1992. Internet Archive.↩︎
If Subject 1 is randomized to sequence $\small{\text{TR}}$, there is not another Subject 1 randomized to sequence $\small{\text{RT}}$. Randomization is not like Schrödinger’s cat.
Hence, the nested term in the guidelines is an insult to the mind.↩︎
Health Canada, HPFB. Guidance for Industry. Conduct and Analysis of Bioavailability and Bioequivalence Studies – Part A: Oral Dosage FormulationsUsed for Systemic Effects. Ottawa. 1992. BEBAC Archive.↩︎
Health Canada, HPFB. Guidance for Industry. Conduct and Analysis of Bioavailability and Bioequivalence Studies – Part B: Oral Modified Release Formulations. Ottawa. 1996. BEBAC Archive.↩︎
WHO Marketing Authorization of Pharmaceutical Products with Special Reference to Multisource (Generic) Products: A Manual for Drug Regulatory Authorities. Geneva. 1998. Internet Archive.↩︎
Elkoshi Z, Behr D, Mirimsky A, Tsvetkov, Danon A. Multiple-Dose Studies can be a More Sensitive Assessment for Bioequivalence than Single-Dose Studies. The Case with Omeprazole. Clin Drug Invest. 2002; 22(9): 585–92. doi:10.2165/00044011-200222090-00003.↩︎
Gleiter CH, Klotz U, Kuhlmann J, Blume H, Stanislaus F, Harder S, Paulus H, Poethko-Müller C, Holz-Slomczyk M. (1998), When Are Bioavailability Studies Required? A German Proposal. J Clin Pharmacol. 1998 38: 904–11. doi:10.1002/j.1552-4604.1998.tb04385.x. Open Access.↩︎
EMEA, CPMP. Note for Guidance on the Investigation of Bioavailability and Bioequivalence. London. 26 July 2001. Online.↩︎
FDA, CDER. Guidance for Industry. Immediate Release Solid Oral Dosage Forms. Scale-Up and Postapproval Changes: Chemistry, Manufacturing, and Controls, In Vitro Dissolution Testing, and In Vivo Bioequivalence Documentation. Rockville. November 1995. Download.↩︎
FDA, CDER. Guidance for Industry. SUPAC-MR: Modified Release SolidOral Dosage Forms. Scale-Up and Postapproval Changes: Chemistry, Manufacturing, and Controls, In Vitro Dissolution Testing, and In Vivo Bioequivalence Documentation. Rockville. September 1997. Download.↩︎
Shah VP, Tsong Y, Sathe P, Liu J-p. In vitro dissolution profile comparison – statistics and analysis of the similarity factor f₂. Pharm Res. 1998; 15: 889–96. doi:10.1023/a:1011976615750.↩︎
Liu J-p, Ma M-C, Chow S-C. Statistical Evaluation of Similarity Factor f₂ as a Criterion for Assessment of Similarity Between Dissolution Profiles. Drug Inf J. 1997; 31: 1255–71. doi:10.1177/009286159703100426.↩︎
Midha KK, Blume HH, editors. Bio-International. Bioavailability, Bioequivalence and Pharmacokinetics. Stuttgart: medpharm; 1993. ISBN 3-88763-019-X.↩︎
Blume HH, Midha KK. Bio-International 92, Conference on Bioavailability, Bioequivalence, and Pharmacokinetic Studies. J Pharm Sci. 1993; 82(11): 1186–9. doi:10.1002/jps.2600821125.↩︎
Simultaneous administration of a stable isotope labelled IV dose would allow to calculate the true clearance in each period. Then it would not be necessary to assume identical clearances in $\small{(3)}$ any more and the problem of highly variable drugs (inflating the CI) could be avoided. However, it would require that the IV formulation is manufactured according to the rules of cGMP and different from the internal standard in MS, which is generally not feasible. Such an approach is only mentioned in Japanese guidelines.↩︎
Blume HH, Midha KK, editors. Bio-International 2. Bioavailability, Bioequivalence and Pharmacokinetic Studies. Stuttgart: medpharm; 1995. ISBN 3-88763-040-8.↩︎
Boddy AW, Snikeris FC, Kringle RO, Wei GCG, Opperman JA, Midha KK. An approach for widening the bioequivalence acceptance limits in the case of highly variable drugs. Pharm Res. 1995; 12(12): 1865–8. doi:10.1023/a:1016219317744.↩︎
Shah VP, Yacobi A, Barr WH, Benet LZ, Breimer D, Dobrinska MR, Endrényi L, Fairweather W, Gillespie W, Gonzalez MA, Hooper J, Jackson A, Lesko LL, Midha KK, Noonan PK, Patnaik R, Williams RL. Workshop Report. Evaluation of Orally Administered Highly Variable Drugs and Drug Formulations. Pharm Res. 1996; 13(11): 1590–4. doi:10.1023/a:1016468018478.↩︎
Schütz H, Labes D, Wolfsegger MJ. Critical Remarks on Reference-Scaled Average Bioequivalence. J Pharm Pharmaceut Sci. 25: 285–96. doi:10.18433/jpps32892.↩︎
EMEA Human Medicines Evaluation Unit / CPMP. Note for Guidance on the Investigation of Bioavailability and Bioequivalence. Draft. London. 17 December 1998.↩︎
Brooks MA, Weifeld RE. A Validation Process for Data from the Analysis of Drugs in Biological Fluids. Drug Devel Ind Pharm. 1985; 11: 1703–28.↩︎
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Buick AR, Doig MV, Jeal SC, Land GS, McDowall RD, Method Validation in the Bioanalytical Laboratory. J Pharm Biomed Anal. 1990; 8(8–12): 629–37. doi:10.1016/0731-7085(90)80093-5. Open Access.↩︎
Karnes ST, Shiu G, Shah VP. Validation of Bioanalytical Methods. Pharm Res. 1991; 8(4): 421–6. doi:10.1023/a:1015882607690.↩︎
AAPS, FDA, FIP, HPB, AOAC. Analytical Methods Validation: Bioavailability, Bioequivalence and Pharmacokinetic Studies. Arlington, VA. December 3–5, 1990.↩︎
Shah VP, Midha KK, Dighe S, McGilveray IJ, Skelly JP, Yacobi A, Layloff T, Viswanathan CT, Cook CE, McDowall RD, Pittman, Spector S. Analytical methods validation: Bioavailability, bioequivalence and pharmacokinetic studies. Eur J Drug Metabol Pharmacokinet. 1991; 16(4): 249–55. doi:10.1007/bf03189968.↩︎
Idanpään-Heikkilä JE. Current Status of the International Comparator Product System for Pharmaceuticals. Presentation at: FIP Bio-International ’99. London. 29 September – 1 October 1999.↩︎
WHO. List of International Comparator products. Geneva. 21 March 2024. Download.↩︎
Siewert M. Innovator Industry Perspective. Presentation at: FIP Bio-International ’99. London. 29 September – 1 October 1999.↩︎
MEB. National recommendations; List of substances exempted from bio-equivalence studies (positive list). The Hague. Internet Archive.↩︎
Danish Medicines Agency. Bioequivalence and labelling of medicinal products with regard to generic substitution. Copenhagen. 13 July 2005, updated 8 February 2010. Internet Archive.↩︎
Shah VP, Midha KK, Findlay JWA, Hill HM, Hulse JD, McGilveray IJ, McKay G, Miller KJ, Patnaik RN, Powell ML, Tonelli A, Viswanathan CT, Yacobi A. Bioanalytical Method Validation – A Revisit with a Decade of Progress. Pharm Res. 2000; 17: 1551–7. doi:10.1023/a:1007669411738↩︎
Viswanathan CT, Bansal S, Booth B, DeStefano AJ, Rose MJ, Sailstad J, Shah VP, Skelly JP, Swann PG, Weiner R. Workshop / Conference Report – Quantitative Bioanalytical Methods Validation and Implementation: Best Practices for Chromatographic and Ligand Binding Assays. AAPS J. 2007; 24(10): 1962–73. doi:10.1007/s11095-007-9291-7.↩︎
Fast DM, Kelley M, Viswanathan CT, O’Shaughnessy J, King SP, Chaudhary A, Weiner R, DeStefano AJ, Tang D. Workshop Report and Follow-Up—AAPS Workshop on Current Topics in GLP Bioanalysis: Assay Reproducibility for Incurred Samples—Implications of Crystal City Recommendations. AAPS J. 2009; 11(2): 238–41. doi:10.1208/s12248-009-9100-9. Free Full Text.↩︎
FDA, CDER, CVM. Guidance for Industry. Bioanalytical Method Validation. Rockville. May 2001. Internet Archive.↩︎
FDA, CDER, , CVM. Guidance for Industry. Bioanalytical Method Validation. Silver Spring. May 2018. Download.↩︎
EMEA, CHMP. Guideline on Validation of Bioanalytical Methods. Draft. London. 19 November 2009. Online.↩︎
EMA, CHMP. Guideline on Validation of Bioanalytical Methods. London. 21 July 2011. Online.↩︎
ICH. Bioanalytical Method Validation And Study Sample Analysis. M10. 22 May 2022. Online.↩︎
EMEA, CHMP. Implementation strategy of ICH Guideline M10 on bioanalytical method validation. Amsterdam. 04 April 2024 . Online.↩︎
FDA, CDER, OGD. Guidance for Industry. Bioequivalence Recommendations for Specific Products. Silver Spring. June 2010. Download.↩︎
EMA, CHMP. Guideline on the pharmacokinetic and clinical evaluation of modified release dosage forms. London. 20 November 2014. Online.↩︎
Schütz H. Primary and secondary PK metrics for evaluation of steady state studies, $\small{C_\text{min}}$ vs. $\small{C_\tau}$, relevance of $\small{C_\text{min}}$/$\small{C_\tau}$ or fluctuation for bioequivalence assessment. Presentation at: 4^th GBHI Workshop. Amsterdam; 12 April 2018. Online.↩︎
Anderson S. Individual Bioequivalence: A problem of Switchability (including Discussions by Metzler CM, Selwyn MR, Liu J-p, Holder DJ, Chinchilli V and a Rejoinder by Anderson S. Biopharm Rep. 1993; 2(2): 1–11. Open Access.↩︎
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Endrényi L. A method for the evaluation of individual bioequivalence. Int J Clin Pharmacol. 1994; 32(9): 497–508. PMID 7820334.↩︎
Esinhart JD, Chinchilli VM. Extension to use of tolerance intervals for the assessment of individual bioequivalence. J Biopharm Stat. 1994; 4: 39–52. doi:10.1080/10543409408835071.↩︎
Chow S-C, Liu J-p. Current issues in bioequivalence trials. Drug Inform J. 1995; 29: 795–804. doi:10.1177/009286159502900302.↩︎
Chen ML. Individual bioequivalence. A regulatory update. J Biopharm Stat. 1997; 7(1): 5–11. doi:10.1080/10543409708835162.↩︎
Hauck WW, Anderson S. Commentary on individual bioequivalence by ML Chen. J Biopharm Stat. 1997; 7(1): 13–6. doi:10.1080/10543409708835163.↩︎
Liu J-p, Chow S-C. Some thoughts on individual bioequivalence. J Biopharm Stat. 1997; 7(1): 41–8. doi:10.1080/10543409708835168.↩︎
Midha KK, Rawson MJ, Hubbard JW. Prescribability and switchability of highly variable drugs and drug products. J Contr Rel. 1999; 62(1-2): 33–40. doi:10.1016/s0168-3659(99)00050-4.↩︎
FDA, CDER. Guidance for Industry. Statistical Approaches to Establishing Bioequivalence. Rockville. January 2001. Download.↩︎
Chow S-C, Shao J, Wang H. Individual bioequivalence testing under 2 × 3 designs. Stat Med. 2002; 21(5): 629–48. doi:10.1002/sim.1056.↩︎
Chow S-C, Liu J-p. Design and Analysis of Bioavailability and Bioequivalence Studies. Boca Raton: Chapman & Hall/CRC Press; 3^rd edition 2009. ISBN 978-1-58488-668-6. p. 596–8. ★↩︎
Hauschke D, Steinijans VW, Pigeot I. Bioequivalence Studies in Drug Development. Methods and Applications. Chichester: Wiley; 2007. ISBN 0-470-09475-3. p. 209. ★↩︎
Benet LZ. Individual Bioequivalence: Have the Opinions of the Scientific Community Changed? In: FDA Advisory Committee for Pharmaceutical Sciences and Clinical Pharmacology Meeting Transcript. US Food and Drug Administration Dockets. Nov 29, 2001. Internet Archive.↩︎
Patterson S. A Review of the Development of Biostatistical Design and Analysis Techniques for Assessing In Vivo Bioequivalence: Part Two. Ind J Pharm Sci. 2001; 63(3): 169–86. Open Access.↩︎
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Schall R, Endrényi L. Bioequivalence: tried and tested. Cardiovasc J Afr. 2010; 21(2): 69–70. PMCID 3721767. Free Full Text.↩︎
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EMA, CHMP. Concept Paper for an Addendum to the Note for Guidance on the Investigation of Bioavailability and Bioequivalence: Evaluation of Bioequivalence of Highly Variable Drugs and Drug Products. London. 27 April 2006. BEBAC Archive.↩︎
EMEA, CHMP. Guideline on the Investigation of Bioequivalence. London. 20 January 2010. Online.↩︎
FDA, OGD. Draft Guidance on Progesterone. Recommended Apr 2010; Revised Feb 2011. Online ↩︎
Davit BM, Chen ML, Conner DP, Haidar SH, Kim S, Lee CH, Lionberger RA, Makhlouf FT, Nwakama PE, Patel DT, Schuirmann DJ, Yu LX. Implementation of a Reference-Scaled Average Bioequivalence Approach for Highly Variable Generic Drug Products by the US Food and Drug Administration. AAPS J. 2012; 14(4): 915–24. doi:10.1208/s12248-012-9406-x. Free Full Text.↩︎
WHO Expert Committee on Specifications for Pharmaceutical Preparations. Multisource (generic) pharmaceutical products: guidelines on registration requirements to establish interchangeability. Fifty-first report. Technical Report Series, No. 992, Annex 6. Geneva. April 2017. Download.↩︎
Health Canada. Guidance Document. Comparative Bioavailability Standards: Formulations Used for Systemic Effects. Ottawa. 2018/06/08. Online.↩︎
Some gastric resistant formulations of diclofenac are HVDPs, practically all topical formulations are HVDPs, whereas diclofenac itself is not a HVD ($\small{CV_\text{w}}$ of a solution ~8%).↩︎
An inglorious counterexample: In the approval of dabigatran (the first univalent direct thrombin (IIa) inhibitor) the originator withheld information about severe bleeding events. Although it is highly variable, reference-scaling is not justified (see NTID III.2–3).↩︎
Note that the model of SABE is based on the true $\small{\sigma_\text{wR}}$, whereas in practice the observed $\small{s_\text{wR}}$ is used. This may lead to a misclassification and thus, and inflated type I error.¹⁰⁸↩︎
FDA, CDER. Guidance for Industry. Bioequivalence Studies With Pharmacokinetic Endpoints for Drugs Submitted Under an ANDA. Draft. Silver Spring. August 2021. Download.↩︎
Tóthfalusi L, Endrényi L, García-Arieta A. Evaluation of bioequivalence for highly variable drugs with scaled average bioequivalence. Clin Pharmacokinet. 2009; 48: 725–43. doi:10.2165/11318040-000000000-00000.↩︎
Picky: $\small{CV_\text{wR}=100\sqrt{\exp(0.294^2)-1}=30.04689\ldots\%}\neq 30\%$!↩︎
Benet L. Why Highly Variable Drugs are Safer. Presentation at: FDA Advisory Committee for Pharmaceutical Science. Rockville. 06 October, 2006. Internet Archive.↩︎
Endrényi L, Tóthfalusi L. Regulatory Conditions for the Determination of Bioequivalence of Highly Variable Drugs. J Pharm Pharmaceut. 2009; 12(1): 138–49. doi:10.18433/j3zw2c. Open Access.↩︎
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Howe WG. Approximate Confidence Limits on the Mean of X+Y Where X and Y are Two Tabled Independent Random Variables. J Am Stat Assoc. 1974; 69(347): 789–94.↩︎
EMA, CHMP PKWP. Questions & Answers: Positions on specific questions addressed to the Pharmacokinetics Working Party. London. 26 January 2011. Online.↩︎
FDA, OGD. Draft Guidance on Warfarin Sodium. Recommended Dec 2012. Download.↩︎
FDA, OGD. Draft Guidance on Dabigatran Etexilate Mesylate. Silver Spring. Recommended Jun 2012; Revised Sept 2015, Jul 2017. Download.↩︎
FDA, OGD. Draft Guidance on Rivaroxaban. Silver Spring. Recommended Sep 2015; Revised Aug 2020. Download.↩︎
FDA, OGD. Draft Guidance on Rivaroxaban. Silver Spring. February 2024. Download.↩︎
FDA, OGD. Draft Guidance on Edoxaban Tosylate. Silver Spring. Recommended May 2017; Revised Mar 2020. Download.↩︎
FDA, CDER. Guidance for Industry. Waiver of In Vivo Bioavailability and Bioequivalence Studies for Immediate-Release Solid Oral Dosage Forms Based on a Biopharmaceutics Classification System. Rockville. August 2000. BEBAC Archive.↩︎
ICH. Biopharmaceutic Classification System-based Biowaivers. M9. Geneva. 20 November 2019. Online.↩︎
Midha KK, McKay G. Bioequivalence; Its History, Practice, and Future. AAPS J. 2009; 11(4): 664–70. doi:10.1208/s12248-009-9142-z. Free Full Text.↩︎
Davit B, Braddy AC, Conner DP, Yu LX. International Guidelines for Bioequivalence of Systemically Available Orally Administered Generic Drug Products: A Survey of Similarities and Differences. AAPS J. 2013; 15(4): 974–90. doi:10.1208/s12248-013-9499-x. Free Full Text.↩︎
Freitas Fernandes EA, van Oudtshoorn J, Tam A, Arévalo González LA, Aurela EG, Potthast H, Mettke K, Kuribayashi R, Shimojo K, Kasuga M, Morales L, Rodríguez Z, Jones B, Ahn C, Yun E, Kim SH, Rodrigues C, Tiong T, Crane C, Walther C, Roost MS, Chen T-L, Hsu L-f, Braddy AC, García-Arieta A, Abalos I, Divinsky M, Alsuwyeh A, Alzenaidy B, Alhar A. The bioequivalence study design recommendations for immediate-release solid oral dosage forms in the international pharmaceutical regulators programme participating regulators and organisations: differences and commonalities. J Pharm Pharmaceut Sci. 2024; 27: 12398. doi:10.3389/jpps.2024.12398.↩︎
WHO / PQT: medicines. Guidance Document: Application of reference-scaled criteria for AUC in bioequivalence studies conducted for submission to PQT/MED. Geneva. 02 July 2021. Online.↩︎
Tam A, García-Arieta A, Abalos I, Freitas Fernandes EA, Mendes Lima Santos G, Rodríguez Martínez Z, Divinsky M, Kariv R, Potthast H, Braddy AC, Rodrigues C, Guzman Aurela E, Arévalo González LC, Gutierrez Triana D, Jones B, Ahn C, Kim H, Kim SH, Kuribayashi R, Myoenzono A, Shimojo K, van Oudtshoorn J, Bigler C, Meincke R, Roost MS, Walther C, Hsu L-f, Crane C, Jarman T. A Survey of the Criteria Used for the Selection of Alternative Comparator Products by Participating Regulators and Organizations of the International Pharmaceutical Regulators Programme. J Pharm Pharmaceut Sci. 2022; 25: 323–39. doi:10.18433/jpps33081.↩︎
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Agency	Uncomplicated drug	HVD(P)	NTID
FDA ¹⁶¹	ABE (any metric) CI \(\small{\subset}\) 80.00–125.00%	RSABE (any metric) \(\small{\textsf{bound}\leq}\) 0 PE \(\small{\subset}\) 80.00–125.00%	RSABE (any metric) \(\small{\textsf{bound}\leq}\) 0 CI \(\small{\subset}\) 80.00–125.00% upper CL of \(\small{\sigma_\text{wT}/\sigma_\text{wR} \leq}\) 2.5
EMA ^{153 131}	ABE (any metric) CI \(\small{\subset}\) 80.00–125.00%	ABEL (\(\small{C_\text{max}}\), \(\small{\textsf{p}AUC}\)) \(\small{uc}\) 50% PE \(\small{\subset}\) 80.00–125.00%	ABE CI \(\small{\subset}\) 90.00–111.11% PSGs: Only for \(\small{AUC}\)
WHO ¹⁵⁶ ¹⁸³	ABE (any metric) CI \(\small{\subset}\) 80.00–125.00%	ABEL (\(\small{C_\text{max}}\), \(\small{AUC}\)) \(\small{uc}\) 50% PE \(\small{\subset}\) 80.00–125.00%	ABE CI \(\small{\subset}\) 90.00–111.11%
HC ¹⁵⁷	ABE \(\small{AUC}\): CI \(\small{\subset}\) 80.0–125.0% \(\small{C_\text{max}}\): PE \(\small{\subset}\) 80.0–125.0%	ABEL (\(\small{AUC}\), \(\small{uc}\) 57.382%) ABE (\(\small{C_\text{max}}\): PE \(\small{\subset}\) 80.0–125.0%)¹⁵⁷	ABE \(\small{AUC}\): CI \(\small{\subset}\) 90.0–112.0% \(\small{C_\text{max}}\): CI \(\small{\subset}\) 80.0–125.0%

A Short History of Bioequivalence

What You Always Wanted to Know
(But Were Afraid to Ask)

Helmut Schütz

January 11, 2025

Introduction

The 1970s

80/20 Rule

95% CI

Westlake’s CI

The Roaring 1980s

75/75 Rule

t-test

ANOVA and beyond

The Boring (?) 1990s

21^st century

PBE/IBE

SABE

RSABE

ABEL

NTIDs

BCS-based Biowaivers

Open Issues

M13

M13A

Implementation Status

Implemented

Implementation process

Not implemented

M13B/C

TODO… (not covered in the article yet)

Acknowledgments

Licenses

Abbreviations

Footnotes and References

A Short History of Bioequivalence

What You Always Wanted to Know(But Were Afraid to Ask)

Helmut Schütz

January 11, 2025

Introduction

The 1970s

80/20 Rule

95% CI

Westlake’s CI

The Roaring 1980s

75/75 Rule

t-test

ANOVA and beyond

The Boring (?) 1990s

21st century

PBE/IBE

SABE

RSABE

ABEL

NTIDs

BCS-based Biowaivers

Open Issues

M13

M13A

Implementation Status

Implemented

Implementation process

Not implemented

M13B/C

TODO… (not covered in the article yet)

Acknowledgments

Licenses

Abbreviations

Footnotes and References

What You Always Wanted to Know
(But Were Afraid to Ask)

21^st century