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Introduction

    

How should we extrapolate the AUC to infinity?

    

Good question because two variants are implemented in software, namely based on the observed (\(\small{C_\textrm{last}}\)) and the predicted (\(\small{\widehat{C}_\textrm{last}}\)) last concentration: \[AUC_{0-\infty}=AUC_{{0-}\textrm{tlast}} + C_\textrm{last}/\widehat{\lambda}_\textrm{z}\tag{1}\] \[AUC_{0-\infty}=AUC_{{0-}\textrm{tlast}} + \widehat{C}_\textrm{last}/\widehat{\lambda}_\textrm{z}\tag{2}\] where \(\small{AUC_{{0-}\textrm{tlast}}}\) is calculated by a trapezoidal rule, \(\small{\widehat{\lambda}_\textrm{z}}\) and \(\small{\widehat{C}_0}\) are the apparent elimination rate constant and the concentration of a virtual intravenous dose at time zero estimated by semilogarithmic regression. Then \[\eqalign{ \widehat{C}_\textrm{last}&=\widehat{C}_0\cdot\exp(-\widehat{\lambda}_\textrm{z} \cdot \textrm{t}_\textrm{last})\\ \small{\textsf{for}}\;\widehat{\lambda}_\textrm{z}&>0}\tag{3}\]

However, this might be a somewhat academic question in most cases.

“When the time course is measured until plasma concentration becomes 5% of its maximum, the relative cutoff errors in AUC, MRT, and VRT are smaller than about 5%, 10%, and 40%, res­pectively […]. If the time course is measured until plasma concentration becomes 1% of its maximum, the relative cutoff errors in AUC, MRT, and VRT are smaller than about 1%, 2%, and 10%, respectively.
Yamaoka et al. (1978)1

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Past and Present

    

In the early days of Noncompartmental Analysis \(\small{(1)}\) was commonly used.

Already 44 years ago \(\small{(2)}\) was occasionally used (e.g., in a project2 sponsored by the FDA) and for the first time re­com­mend­ed 33 years ago.3 4 5 6 For a while it was mandatory for manuscripts submitted to the ‘Bio­equi­va­lence Sec­tion’ of the Inter­na­tio­nal Jour­nal of Clinical Pharmacology, Therapy and Toxicology (edited by one of the pioneers of bioequivalence, Henning Blume).
Other references: By an author of the FDA7 and more recent ones as well.8 9 10 11

    

Makes sense. \(\small{C_\textrm{last}}\) might be close to the Lower Limit of Quantification (LLOQ) and hence, has the lowest accuracy and highest variability of the entire profile. If one bases the extrapolation on it, one sets a fox to keep the geese. The intrinsic high variability of \(\small{C_\textrm{last}}\) propagates into the extrapolated area. Contrary to that, \(\small{\widehat{C}_\textrm{last}}\) takes information of preceding (higher, and therefore, more accurate and precise) concentrations into account and thus, is in general more reliable.

“In general, we recommend use of the predicted rather than the observed last concentration when computing the extrapolated area.
Gabrielsson and Weiner (2016)11

Little is mentioned in guidelines. According to the WHO, the method has to be specified in the protocol.12 The FDA,13 China’s CDE, the ANVISA,14 and the WHO15 recommend \(\small{(1)}\), whereas Health Canada16 recommends \(\small{(2)}\). Nothing is stated by the EMA17 18 and the ICH.19

I’m using solely \(\small{(2)}\) since the early 1990s and never received a single deficiency letter in this respect (yes, none from the FDA as well). Of course, the method should be outlined in the protocol.16 Referring only to an SOP is not a good idea because it is not accessible to regulatory assessors in the first place. My standard wording for drugs in IR formulations following monoexponential elimination:

“Extrapolation from \(\small{\textrm{t}_\textrm{last}}\) to infinite time will be performed by fitting – at least three – terminal concentrations to the monoexponential model \(\small{\widehat{C}_\textrm{t}=\widehat{C}_0\cdot\exp(-\widehat{\lambda}_\textrm{z}\cdot \textrm{t})}\) by means of un­weight­ed semilogarithmic regression. The starting point for the estimation will default to \(\small{\geq 2\times \textrm{t}_\textrm{max}}\).20 The selected time points will be adjusted if deemed necessary, i.e., a visual inspection of fits will be per­formed.3 5 8 20 Increasing concentrations in the late phase of the profile will be excluded from the estimation of \(\small{\widehat{\lambda}_\textrm{z}}\). To avoid under- or over-esti­ma­tion of the extrapolated area, the estimated last concentration \(\small{\widehat{C}_\textrm{last}}\) will be used in extra­polations.3 5 8 11

If PK is expected to be multiphasic, i.e., with distribution phase(s) and elimination, the second sentence is:

“The intersection of the last two phase lines in the (semilogarithmic) profile post \(\small{C_\textrm{max}}\) is used as a ‘visual marker’ for the beginning of the monoexponential terminal phase.20

    

Algorithms for the estimation of \(\small{\widehat{\lambda}_\textrm{z}}\) will be ela­borated in another article.

Dr. David Bowman inside HAL 9000

It must be mentioned that any algorithm might fail on ‘flat’ profiles (con­trolled re­lease formulations with flip-flop PK) or on profiles of multiphasic release formulations.

Nota bene, visual inspection of fits is man­da­tory.3 5 8 20 It was shown in simulations of two-com­part­ment models to outperform standard automatic methods.21 Our brain is an excellent pattern rec­og­​ni­tion machine.

One software vendor rightly notes:

“Using this methodology, Phoenix will almost always compute an estimate for Lambda Z. It is the user’s responsibility to evaluate the appropriateness of the estimated value.
Certara (2020)22
(my emphasis)

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Software

Both methods are implemented in numerous pieces of software. Most commonly used terminology:

Term Metric
AUClast \(\small{AUC_{{0-}\textrm{tlast}}}\)
AUCINF_obs \(\small{AUC_{0-\infty}}\) based on \(\small{(1)}\)
AUCINF_pred \(\small{AUC_{0-\infty}}\) based on \(\small{(2)}\)
Clast_pred \(\small{\widehat{C}_\textrm{last}}\)
Lambda_z \(\small{\widehat{\lambda}_\textrm{z}}\)
Lambda_z_intercept \(\small{\widehat{C}_0}\)

Already in 1993 both methods were implemented in TopFit 2.0.23 They are available in Phoenix WinNonlin24 (see the User’s Guide or the online manual25 for details), PK­ana­lix,26 the R packages PKNCA,27 28 ncappc,29 qpNCA,30 and ubiquity,31 32 33 as well as in the Julia library Pumas.34 35

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Simulation

    

One-compartment model, first order absorption and elimination, no lag time. PK-parameters: \(\small{D=200}\), \(\small{f=0.80}\), \(\small{V=3}\), absorption half life 45 minutes, elimination half life five hours.
The error distribution was log-normal, where the error increased with decreasing concentrations. The LLOQ was set to 4% of the model’s \(\small{C_\textrm{max}}\). Simulated concentrations below the LLOQ prior to \(\small{t_\textrm{max}}\) were set to zero, and later ones to NA. Automatic algorithm for the estimation of \(\small{\widehat{\lambda}_\textrm{z}}\) based on maximizing \(\small{R_\textrm{adj}^2}\), which is the default in all other software as well.

    

Sorry, 264 LOC.

Function Purpose
one.comp() Calculates concentrations, the exact (i.e., the model’s) \(\small{C_\textrm{max}}\), \(\small{t_\textrm{max}}\), \(\small{AUC_{0-\textrm{tlast}}}\), \(\small{AUC_{0-\infty}}\), and LLOQ for given vector of sampling times, PK-parameters, and LLOQ specified as a fraction of the model’s \(\small{C_\textrm{max}}\).
micro2macro() Converts the model’s PK-parameters (micro constants \(\small{f}\), \(\small{D}\), \(\small{V}\), \(\small{k_{01}}\), and \(\small{k_{10}}\)) to macro (hybrid) constants \(\small{A}\), \(\small{B}\), \(\small{\alpha}\), and \(\small{\beta}\). Note that in a model without lag time \(\small{\left| A \right| = B}\).
round.up() Rounds x up to the next multiple of y.
sampling() Calculates a vector of ‘optimal’ sampling times: Equally spaced to \(\small{t_\textrm{max}}\) and then following a geometric progression to \(\small{t_{\textrm{tlast}}}\).
est.elim() Estimates the apparent elimination by semilogarithmc regression. Returns \(\small{R_\textrm{adj}^2}\), \(\small{\widehat{C}_0}\), \(\small{\widehat{\lambda}_\textrm{z}}\), start- and end-times, and the number of values used. If estimation fails (slope ≥ 0), returns NAfor all.
calc.AUC() Calculates \(\small{AUC_{{0-}\textrm{tlast}}}\) by a trapezoidal rule. Implemented are linlog (default) and linear. See this article for details.
AUC.extr() Extrapolates the \(\small{AUC}\). Returns \(\small{AUC_{{0-}\textrm{tlast}}}\) (for comparison), \(\small{AUC_{0-\infty}}\) (observed and pre­dict­ed), \(\small{C_\textrm{last}}\), and \(\small{\widehat{C}_\textrm{last}}\).
sum.simple() Nonparametric summary (i.e., removes the arithmetic mean from summary().
geom.stat() Calculates the geometric mean and CV. 
sim.auc <- function(D, f, V, t12.a, t12.e, n1, n2, tlast,
                    mins, CV0, LLOQ.f, rule, nsims, setseed = TRUE,
                    show.fits = FALSE, progress = FALSE) {
  one.comp <- function(f, D, V, k01, k10, t, LLOQ.f) {
    # one-compartment model, first order absorption
    # and elimination, no lagtime
    x <- micro2macro(f, D, V, k01, k10) # get hybrid (macro) constants
    if (!isTRUE(all.equal(k01, k10))) { # common: k01 != k10
      C    <- f * D * k01 / (V * (k01 - k10)) *
              (exp(-k10 * t) - exp(-k01 * t))
      # values based on the model
      tmax <- log(k01 / k10) / (k01 - k10)
      Cmax <- f * D * k01 / (V * (k01 - k10)) *
              (exp(-k10 * tmax) - exp(-k01 * tmax))
      AUC  <- f * D / V / k10
      AUCt <- (x$C[["A"]] - x$C[["A"]] * exp(-x$E[["alpha"]] * tlast)) /
              x$E[["alpha"]] +
              (x$C[["B"]] - x$C[["B"]] * exp(-x$E[["beta"]] * tlast)) /
              x$E[["beta"]]
    } else {                            # flip-flop
      k    <- k10
      C    <- f * D / V * k * t * exp(-k * t)
      tmax <- 1 / k
      Cmax <- f * D / V * k * tmax * exp(-k * tmax)
      AUC  <- f * D / V / k
      AUCt <- NA # sorry, no idea
    }
    LLOQ <- Cmax * LLOQ.f
    C[C <= LLOQ] <- NA                # set values below the LLOQ to NA
    C[which(is.na(C[t < tmax]))] <- 0 # set NAs prior tmax to zero
    res <- list(C = C, Cmax = Cmax, tmax = tmax, AUC = AUC, AUCt = AUCt,
                LLOQ = LLOQ)
    return(res)
  }

  micro2macro <- function(f, D, V, k01, k10) {
    # Convert parameters (micro constants) to macro (hybrid) constants
    # Coefficients (C) and exponents (E)
    C     <- f * D * k01 / (V * (k01 - k10))
    C     <- setNames(c(-C, +C), c("A", "B"))
    E     <- setNames(c(k01, k10), c("alpha", "beta"))
    macro <- list(C = C, E = E)
    return(macro)
  }

  round.up <- function(x, y) {
    # round x up to the next multiple of y
    return(y * (x %/% y + as.logical(x %% y)))
  }

  sampling <- function(f, D, V, k01, k10, n1, n2, tlast, mins, LLOQ.f) {
    # <= tmax: equally spaced
    # >= tmax: geometric progression
    # rounded to 'mins' minutes
    tmax <- one.comp(f, D, V, k01, k10, t = tlast, LLOQ.f)$tmax
    t    <- seq(0, tmax, length.out = n1)
    for (i in (n1+1):(n1+n2-1)) {
      t[i] <- t[i-1] * (tlast / tmax)^(1 / (n2 - 1))
    }
    t <- round.up(t * 60, mins) / 60
    t[length(t)] <- tlast
    return(t)
  }

  est.elim <- function(t, C) {
    # estimate lambda.z by "maximum adjusted R-squared approach"
    data   <- data.frame(t = t, C = C)
    Cmax   <- max(data$C, na.rm = TRUE)
    tmax   <- data$t[data$C[!is.na(data$C)] == Cmax]
    data   <- data[data$t > tmax, ]        # discard tmax and earlier
    data   <- data[complete.cases(data), ] # discard NAs
    lz.end <- tail(data$t, 1)
    # start with the last three concentrations
    x      <- tail(data, 3)
    r2     <- a <- b <- numeric()
    m      <- lm(log(C) ~ t, data = x)
    a[1]   <- coef(m)[[1]]
    b[1]   <- coef(m)[[2]]
    r2[1]  <- summary(m)$adj.r.squared
    # work backwards towards tmax
    i      <- 1
    for (j in 4:nrow(data)) {
      i     <- i + 1
      x     <- tail(data, j)
      m     <- lm(log(C) ~ t, data = x)
      a[i]  <- coef(m)[[1]]
      b[i]  <- coef(m)[[2]]
      r2[i] <- summary(m)$adj.r.squared
      # don't proceed if no improvement
      if (r2[i] < r2[i-1] | abs(r2[i] - r2[i-1]) < 0.0001) break
    }
    # location of the largest adjusted R2
    loc <- which(r2 == max(r2, na.rm = TRUE))[1]
    if (b[loc] >= 0 || r2[loc] <= 0) { # not meaningful
      R2adj <- intcpt <- lambda.z <- lz.start <- lz.end <- lz.n <- NA
    } else {
      R2adj    <- r2[loc]
      intcpt   <- a[loc]
      lambda.z <- -b[loc]
      lz.start <- x$t[2]
      lz.n     <- nrow(x)
    }
    res <- data.frame(R2adj = R2adj, intcpt = intcpt, lambda.z = lambda.z,
                      lz.start = lz.start, lz.end = lz.end, lz.n = lz.n)
    return(res)
  }

  calc.AUC <- function(t, C, rule = "linlog",
                       digits = 5, details = FALSE) {
    x <- data.frame(t = t, C = C, pAUC = 0)
    x <- x[with(x, order(t)), ]
    x <- x[complete.cases(x), ]
    for (i in 1:(nrow(x) - 1)) {
      if (rule == "linlog") { # linear-up / log-down
        if (x$C[i+1]  < x$C[i]) {
          x$pAUC[i+1] <- (x$t[i+1] - x$t[i]) * (x$C[i+1] - x$C[i]) /
                         log(x$C[i+1] / x$C[i])
        } else {
          x$pAUC[i+1] <- 0.5 * (x$t[i+1] - x$t[i]) *
                         (x$C[i+1] + x$C[i])
        }
      } else {               # linear
        x$pAUC[i+1] <- 0.5 * (x$t[i+1] - x$t[i]) *
                       (x$C[i+1] + x$C[i])
      }
    }
    x$AUC <- cumsum(x$pAUC)
    x     <- x[with(x, order(t)), ] # sort by time
    if (details) {                  # entire data.frame
      res <- round(x, digits)
    } else {                        # only tlast and AUClast
      res <- setNames(as.numeric(round(tail(x[, c(1, 4)], 1), digits)),
                      c("tlast", "AUClast"))
    }
    if (rule == "linlog") {         # cosmetics ;-)
      attr(res, "trapezoidal rule") <- "linear-up/log-down"
    } else {
      attr(res, "trapezoidal rule") <- "linear"
    }
    return(res)
  }

  AUC.extr <- function(t, C, intcpt, lambda.z) {
    x           <- calc.AUC(t, C, rule = rule, details = TRUE)
    AUClast     <- tail(x$AUC, 1)
    Clast_obs   <- tail(x$C, 1)
    AUCinf_obs  <- AUClast + tail(x$C, 1) / lambda.z
    Clast_pred  <- exp(intcpt - lambda.z * tail(x$t, 1))
    AUCinf_pred <-  AUClast + Clast_pred / lambda.z
    res <- list(AUClast = AUClast, Clast_obs = Clast_obs,
                AUCinf_obs = AUCinf_obs, Clast_pred = Clast_pred,
                AUCinf_pred = AUCinf_pred)
    return(res)
  }

  sum.simple <- function(x, digits = 4) {
    # nonparametric summary: remove arithmetic means
    res <- summary(x)[-4, ]
    return(res)
  }

  geom.stat <- function(x) {
    stats <- as.data.frame(matrix(nrow = 2, ncol = ncol(x),
                                  dimnames = list(c("geom. mean",
                                                    "geom. CV (%)"),
                                                    names(x))))
    for (i in 1:ncol(stats)) {
      stats[1, i] <- exp(mean(log(x[, i]), na.rm = TRUE))
      stats[2, i] <- 100 * sqrt(exp(sd(log(x[, i]), na.rm = TRUE)^2) - 1)
    }
    return(stats)
  }

  k01    <- log(2) / t12.a   # absorption rate constant
  k10    <- log(2) / t12.e   # elimination rate constant
  # generate sampling schedule
  t      <- sampling(f, D, V, k01, k10, n1, n2, tlast, mins, LLOQ.f)
  x      <- one.comp(f, D, V, k01, k10, t, LLOQ.f)
  C0     <- x$C              # theoretical profile (based on model)
  Cmax   <- x$Cmax
  tmax   <- x$tmax
  LLOQ   <- x$LLOQ
  CV     <- CV0 - C0 * 0.006 # variability increases with decreasing C
  varlog <- log(CV^2 + 1)
  sdlog  <- sqrt(varlog)
  aggr1  <- data.frame(R2adj = NA_real_, intcpt = NA_real_,
                       lambda.z = NA_real_, RE = NA_real_, t.half = NA_real_,
                       lz.start = NA_real_, lz.end = NA_real_,
                       lz.n = NA_integer_)
  aggr2  <- data.frame(AUClast = NA_real_, Clast_obs = NA_real_,
                       AUCinf_obs = NA_real_, Clast_pred = NA_real_,
                       AUCinf_pred = NA_real_)
  aggr3  <- data.frame(matrix(NA, nrow = nsims, ncol = length(t)))
  C      <- numeric()
  if (progress) pb <- txtProgressBar(0, 1, 0, width = NA, style = 3)
  if (setseed) set.seed(123456)
  for (i in 1:nsims) {
    for (j in 1:length(C0)) {
      if (is.na(C0[j])) {
        C[j] <- NA # otherwise, rlnorm() fails
      } else {
        C[j] <- rlnorm(1, meanlog = log(C0[j]) - 0.5 * varlog[j],
                       sdlog = sdlog[j])
      }
    }
    # C < LLOQ set to NA, NAs prior to tmax set to zero
    Cmax.tmp <- max(C, na.rm = TRUE)
    tmax.tmp <- t[C[!is.na(C)] == Cmax.tmp]
    C[C <= LLOQ] <- NA
    C[which(is.na(C[t < tmax.tmp]))] <- 0
    aggr3[i, ] <- C
    aggr1[i, c(1:3, 6:8)] <- est.elim(t, C)[1, ]
    if (!is.na(aggr1$lambda.z[i])) { # only if successful fit
      aggr1$t.half[i] <- log(2) / aggr1$lambda.z[i]
      aggr1$RE[i]     <- 100 * (aggr1$lambda.z[i] - k10) / k10
      aggr2[i, ]      <- as.numeric(AUC.extr(t, C,
                                             aggr1$intcpt[i],
                                             aggr1$lambda.z[i]))
    }
    if (progress) setTxtProgressBar(pb, i / nsims)
  }
  if (progress) close(pb)
  method <- "trapezoidal rule"
  ifelse (rule == "linlog",
    method <- paste("linear-up / log-down", method),
    method <- paste("linear", method))
  NAs <- data.frame(time = t,
                    NAs = sapply(aggr3, function(x) sum(length(which(is.na(x))))))
  cat("Results from the model (without error)",
      "\n  AUClast", sprintf("%.3f", x$AUCt),
      "\n  AUCinf ", sprintf("%.3f", x$AUC),
      paste0("\n\n", prettyNum(nsims, big.mark = ","),
             " simulated profiles, ", method, "\n"))
  print(NAs, row.names = FALSE)
  if (nrow(sum.simple(aggr1)) == 6) { # there is a NA row
    failed <- as.integer(gsub("[^[:digit:]]", "", sum.simple(aggr1)[6, 1]))
    cat("In", failed, "profile(s) automatic estimation of the",
        "\napparent elimination failed (nonnegative slope).\n\n")
  }
  print(signif(geom.stat(aggr2), 4))
  if (show.fits) {
    cat("\n")
    print(sum.simple(aggr1))
  }
}

D         <- 200      # dose
f         <- 0.8      # fraction absorbed (BA)
V         <- 3        # volume of distribution
t12.a     <- 0.75     # absorption half life
t12.e     <- 5        # elimination half life
n1        <- 4        # number of samples <= tmax
n2        <- 9        # number of samples >= tmax (n = n1 + n2 - 1)
tlast     <- 24       # last sampling time point (h)
mins      <- 15       # rounding of sampling times (minutes)
CV0       <- 0.40     # CV at low concentrations
LLOQ.f    <- 0.04     # fraction of theoretical Cmax
rule      <- "linlog" # if you insist: "linear"
nsims     <- 2.5e3L   # number of simulations
setseed   <- TRUE     # for reproducibility (default)
show.fits <- FALSE    # TRUE for nonparametric summary of fits
progress  <- FALSE    # TRUE for a progress bar
sim.auc(D, f, V, t12.a, t12.e, n1, n2, tlast, mins, CV0, LLOQ.f,
        rule, nsims, setseed, show.fits, progress)
# Results from the model (without error) 
#   AUClast 368.471 
#   AUCinf  384.719 
# 
# 2,500 simulated profiles, linear-up / log-down trapezoidal rule
#   time NAs
#   0.00   0
#   1.00   0
#   1.75   0
#   2.50   0
#   3.25   0
#   4.50   0
#   5.75   0
#   7.75   0
#  10.25   0
#  13.75   0
#  18.25   2
#  24.00 531
# In 6 profile(s) automatic estimation of the 
# apparent elimination failed (nonnegative slope).
# 
#              AUClast Clast_obs AUCinf_obs Clast_pred AUCinf_pred
# geom. mean   356.700     2.769    382.000      2.645     380.800
# geom. CV (%)   8.316    41.520      9.641     39.570       9.609
    

Even the linear-up/log-down underestimates \(\small{AUC_{{0-}\textrm{tlast}}}\) because the profile is concave up to the inflection point at \(\small{2\times t_\textrm{max}}\). There is nothing we can do about (see another article).

    

Note that the variability of \(\small{C_\textrm{last}}\) is larger than the one of \(\small{\widehat{C}_\textrm{last}}\). As expected, and it propagates partly to the extra­polated \(\small{AUC\textrm{s}}\), where the variability based on \(\small{(1)}\) is – given, slightly – larger then the one based on \(\small{(2)}\).

In either case the variability of \(\small{AUC_{0-\infty}}\) is larger than the one of \(\small{AUC_{{0-}\textrm{tlast}}}\). For immediate release formulations point estimates of \(\small{\textrm{p}AUC_{0-\textrm{t}\,\geq\,\small{2\times t_\textrm{max}}}}\)  are stable, only variability of the respective partial areas increases with time.36

“Once absorption is over, formulation differences no longer apply.
Midha et al. (1996)36

However, this might be relevant in jurisdictions applying Average Bio­equi­va­lence with Ex­pand­ing Limits (ABEL) and \(\small{AUC_{0-\infty}}\) is a primary pharmacokinetic metric because the late part of the profile represents absorption (e.g., controlled release formulations for the EMA, where reference-scaling is only acceptable for \(\small{C_\textrm{max}}\) and the sample size depends on \(\small{AUC}\); see this article for an example).

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Conclusion

\(\small{\widehat{\lambda}_\textrm{z}}\) obtained by an automatic algorithm should be inspected for its plausibility.

\(\small{AUC_{0-\infty}}\) based on either \(\small{(1)}\) or \(\small{(2)}\) can be used. The planned method must be unambiguous stated in the protocol.

From a theoretical point of view – and as demonstrated in the Simulation above – \(\small{AUC_{0-\infty}}\) calculated by \(\small{(2)}\) is expected to result in slighty lower variability and therefore, is recommended.2 3 4 5 6 7 8 9 10 11 16 However, in the com­mon case that the sample size is based on \(\small{C_\textrm{max}}\),37 this advantage over \(\small{(1)}\) likely is not relevant.

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Licenses

CC BY 4.0 Helmut Schütz 2024
R GPL 3.0, klippy MIT, pandoc GPL 2.0.
1st version March 6, 2022. Rendered May 12, 2024 01:05 CEST by rmarkdown via pandoc in 0.18 seconds.

Footnotes and References

  1. Yamaoka K, Nakagawa T, Uno T. Statistical Moments in Pharmacokinetics. J Pharmacokin Biopharm. 1978; 6(6): 547–58. doi:10.1007/bf01062109.↩︎

  2. Upton RA, Sansom L, Guentert TW, Powell JR, Thiercellin J-F, Shah VP, Coates PE, Riegelman S. Evalu­a­tion of the Ab­sorption from 15 Commercial Theophylline Products Indicating Deficiencies in Currently Ap­plied Bio­avail­abi­lity Criteria. J Pharmacokin Biopharm. 1980; 8(3): 229–42. doi:10.1007/BF01059644.↩︎

  3. Schulz H-U, Steinijans, VW. Striving for standards in bioequivalence assessment: a review. Int J Clin Pharm Ther Toxicol. 1991; 29(8): 293–8. PMID:1743802.↩︎

  4. Purves RD. Bias and Variance of Extrapolated Tails for Area-Under-the-Curve (AUC) and Area-Under-the-Moment-Curve (AUMC). J Pharmacokin Biopharm. 1992; 20(5): 501–10. doi:10.1007/BF01061468.↩︎

  5. Sauter R, Steinijans VW, Diletti E, Böhm E, Schulz H-U. Presentation of results from bioequivalence studies. Int J Clin Pharm Ther Toxicol. 1992; 30(7): 233–56. PMID:1506127.↩︎

  6. Wagner JG. Pharmacokinetics for the Pharmaceutical Scientist. Lancaster, Basel: Technomic; 1993. p. 88.↩︎

  7. Abdallah HY. An area correction method to reduce intrasubject variability in bioequivalence studies. J Pharm Phar­ma­ceut Sci. 1998; 1(2): 60–5. Open Access Open Access.↩︎

  8. Hauschke D, Steinjans V, Pigeot I. Bioequivalence Studies in Drug Development. Chichester: Wiley; 2007. p. 131.↩︎

  9. Derendorf H, Gramatté T, Schäfer HG, Staab A. Pharmakokinetik kompakt. Grundlagen und Praxisrelevanz. Stuttgart: Wissenschaftliche Verlagsanstalt; 3. Auflage 2011. p. 127. [German]↩︎

  10. Fisher D, Kramer W, Burmeister Getz E. Evaluation of a Scenario in Which Estimates of Bioequivalence Are Biased and a Proposed Solution: tlast (Common). J Clin Pharm. 2016; 56(7): 794–800. doi:10.1002/jcph.663. Open Access Open Access.↩︎

  11. Gabrielsson J, Weiner D. Pharmacokinetic and Pharmacodynamic Data Analysis. Stockholm: Apotekar­so­cie­te­ten; 5th ed. 2016. p. 147.↩︎

  12. WHO. Technical Report Series, No. 996. Annex 9. Guidance for organizations performing in vivo bioequivalence studies. Geneva. 2016. Online.↩︎

  13. FDA, CDER. Draft Guidance. Bioequivalence Studies With Pharmacokinetic End­points for Drugs Submitted Under an ANDA. Silver Spring. August 2021. Download.↩︎

  14. ANVISA. Resolução - RDC Nº 742. Dispõe sobre os critérios para a condução de estudos de biodisponibilidade re­la­tiva / bio­equivalência (BD/BE) e estudos farmacocinéticos. Brasilia. August 10, 2022. Effective July 3, 2023. Online.↩︎

  15. WHO. Technical Report Series, No. 1003. Annex 6. Multisource (generic) pharmaceutical products: guidelines on registration requirements to establish interchangeability. Section 7.4.7 Parameters to be assessed. Geneva. 2017. Online.↩︎

  16. Health Canada. Guidance Document: Conduct and Analysis of Comparative Bioavailability Studies. Appen­dix 1. Ottawa. 2018/06/08. Online.↩︎

  17. EMA, CHMP. Guideline on the Investigation of Bioequivalence. London. 20 January 2010. Online.↩︎

  18. EMA, CHMP. Guideline on the pharmacokinetic and clinical evaluation of modified release dosage forms. London. 20 November 2014. Online.↩︎

  19. ICH. Bioequivalence for Immediate Release Solid Oral Dosage Forms. M13A. Draft version. 20 De­cem­ber 2022. Online.↩︎

  20. Scheerans C, Derendorf H, Kloft C. Proposal for a Standardised Identification of the Mono-Exponential Ter­mi­nal Phase for Orally Administered Drugs. Biopharm Drug Dispos. 2008; 29(3): 145–57. doi:10.1002/bdd.596.↩︎

  21. Noe DA. Performance characteristics of the adjusted r2 algorithm for determining the start of the terminal disposition phase and comparison with a simple r2 algorithm and a visual inspection method. Pharmaceut Stat. 2019; 1–13. doi:10.1002/pst.1979.↩︎

  22. Certara USA, Inc. Princeton, NJ. 7/9/20. Lambda Z or Slope Estimation settings. Online.↩︎

  23. Heinzel G, Woloszczak R, Thomann R. TopFit 2.0. Pharmacokinetic and Pharmacodynamic Data Analysis Sys­tem for the PC. Stuttgart: Springer; 1993.↩︎

  24. Certara USA, Inc. Princeton, NJ. 2022. Phoenix WinNonlin. Online.↩︎

  25. Certara USA, Inc. Princeton, NJ. 7/9/20. NCA parameter formulas. Online.↩︎

  26. LIXOFT, Antony, France. 2021. PKanalix Documentation. NCA parameters. Online.↩︎

  27. Denney W, Duvvuri S, Buckeridge C. Simple, Automatic Noncompartmental Analysis: The PKNCA R Package. J Phar­ma­cokinet Pharmacodyn. 2015; 42(1): 11–107, S65. doi:10.1007/s10928-015-9432-2.↩︎

  28. Denney B, Buckeridge C, Duvvuri S. PKNCA. Perform Pharmacokinetic Non-Com­part­mental Analysis. Pack­age version 0.10.2. 2023-04-29. CRAN.↩︎

  29. Acharya C, Hooker AC, Turkyilmaz GY, Jonsson S, Karlsson MO. ncappc: NCA Calculations and Popu­la­tion Model Dia­gnosis. Pack­age version 0.3.0. 2018-08-24. CRAN.↩︎

  30. Huisman J, Jolling K, Mehta K, Bergsma T. qpNCA: Noncompartmental Pharmacokinetic Analysis by qPhar­metra. Pack­age version 1.1.6. 2021-08-16. CRAN.↩︎

  31. Harrold JM, Abraham AK. Ubiquity: a framework for physiological/mechanism-based pharma­co­kinetic / pharma­co­dynamic model development and deployment. J Pharmacokinet Pharmacodyn. 2014; 41(2), 141–51. doi:10.1007/s10928-014-9352-6.↩︎

  32. Harrold J. ubiquity: PKPD, PBPK, and Systems Pharmacology Modeling Tools. Pack­age version 2.0.3. 2024-03-08. CRAN.↩︎

  33. Harrold J. Noncompartmental Analysis. 2021-09-03. CRAN.↩︎

  34. Rackauckas C, Ma Y, Noack A, Dixit V, Kofod Mogensen P, Byrne S, Maddhashiya S, Bayoán Santiago Cal­de­rón J, Ny­berg J, Gobburu JVS, Ivaturi V. Accelerated Predictive Healthcare Analytics with Pumas, a High Per­for­mance Phar­ma­ceutical Modeling and Simulation Platform. Nov 30, 2020. doi:10.1101/2020.11.28.402297. Pre­print: bioRxiv 2020.11.28.40229.↩︎

  35. Pumas-AI, Inc. 2022. The NCA Functions. Online.↩︎

  36. Midha KK, Hubbard JW, Rawson MJ. Retrospective evaluation of relative extent of absorption by the use of partial areas under plasma concentration versus time curves in bioequivalence studies on conventional release products. Eur J Pharm Sci. 1996; 4(6): 381–4. doi:10.1016/0928-0987(95)00166-2.↩︎

  37. Very rarely the variability of \(\small{AUC}\) is larger than the one of \(\small{C_\textrm{max}}\) and hence, drives the sample size.↩︎