Bioequivalence and Bioavailability Forum

Hi arl_stat,

❝ […] achievement of Steady state using NOSTASOT (Non-Statistical-significance–of-Trend) method.

Never heard of this abbreviation. THX for the explanation.

❝ Please help me out for SAS codes for analysis.

Sorry, I’m not equipped with ‘’.
Easy in any software. Run a linear regression of pre-dose concentrations vs time and test the slope against zero (or whether zero is included in the 95% CI of the slope). The former should be part of the output.
However, I don’t recommend it (see here and there). The EMA stated:*

Achievement of steady state can be evaluated by collecting pre-dose samples on the day before the PK assessment day and on the PK assessment day. A specific statistical method to assure that steady state has been reached is not considered necessary in bioequivalence studies. Descriptive data is sufficient.

• Regression approaches lead into trouble.
  • If you have a small within-subject variability of pre-dose concentrations, possibly the slope will significantly differ from zero in some subjects. You will conclude that steady state is not reached and exclude those subjects.
    
  • If you have a large within-subject variability of pre-dose concentrations, even a – visually – obvious positive slope will not significantly differ from zero. You will conclude that steady state is reached although the saturation is not complete.
    
  • An example in at the end. Same saturation, only different variabilities of k₁₀.
• AFAIK, not a regulatory requirement anywhere.

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Simulated one-compartment model: V = 4, D = 500, k₀₁ = 0.6931472 h^–1, k₁₀ = 0.0504107 h^–1, τ = 24. Sufficient built-up of (pseudo-) state state: 96 h = 6.98 half lives.

n    <- 3 # number of pre-dose samples for the regression
data <- data.frame(dose = 2:6, t = seq(-96, 0, 24))
# low and high variability of pre-dose concentrations
lo   <- cbind(data, C = c(40.08, 51.87, 55.77, 56.64, 57.66))
hi   <- cbind(data, C = c(40.41, 52.03, 55.36, 56.74, 57.25))
res  <- data.frame(variability = c("low", "high"), int =  NA_real_,
                   slope = NA_real_, signif = "no   ",
                   CL.lo = NA_real_, CL.hi = NA_real_)
for (j in 1:nrow(res)) {
  if (j == 1) {
    tmp <- tail(lo, n)
  } else {
    tmp <- tail(hi, n)
  }
  muddle       <- lm(C ~ t, data = tmp)        # linear regression
  res$int[j]   <- signif(coef(muddle)[[1]], 5) # intercept
  res$slope[j] <- signif(coef(muddle)[[2]], 5) # slope
  if (anova(muddle)[1, 5] < 0.05) res$signif[j] <- "yes  "
  res[j, 5:6]  <- sprintf("%+.6f", confint(muddle, level = 0.95)[2, ])
}
names(res)[4] <- "signif # 0?"
print(lo, row.names = FALSE)
print(hi, row.names = FALSE)
print(res, row.names = FALSE)
op <- par(no.readonly = TRUE)
par(mar = c(4, 4, 2.5, 0.5))
split.screen(c(2, 1))
screen(1) # saturation phase
plot(lo$t, lo$C,  type = "n", axes = FALSE,
     xlab = "time", ylab = "concentration",
     ylim = range(c(lo$C, hi$C)))
grid(nx = NA, ny = NULL); box()
abline(v = unique(lo$t), lty = 3, col = "lightgrey")
lines(lo$t, lo$C, col = "blue", lwd = 2)
points(lo$t, lo$C, pch = 19, col = "blue", cex = 1.5)
lines(hi$t, hi$C, col = "red", lwd = 2)
points(hi$t, hi$C, pch = 19, col = "red", cex = 1.5)
axis(1, at = lo$t)
axis(2, las = 1)
axis(3, at = unique(lo$t),
     label = paste0("dose #", unique(lo$dose)))
screen(2) # last 3 pre-dose concentrations
plot(tail(lo$t, n), tail(lo$C, n), type = "n", axes = FALSE,
     xlab = "time", ylab = "concentration",
     ylim = range(c(tail(lo$C, n), tail(hi$C, n))))
grid(nx = NA, ny = NULL); box()
abline(v = unique(tail(lo$t, n)), lty = 3, col = "lightgrey")
lines(tail(lo$t, n), tail(lo$C, n), col = "blue", lwd = 2)
lines(tail(hi$t, n), tail(hi$C, n), col = "red", lwd = 2)
segments(x0 = -48, y0 = res$int[1] - res$slope[1] *48,
         x1 = 0, y1 = res$int[1], col = "blue", lty = 2)
segments(x0 = -48, y0 = res$int[2] - res$slope[2] *48,
         x1 = 0, y1 = res$int[2], col = "red", lty = 2)
axis(1, at = lo$t)
axis(2, las = 1)
axis(3, at = unique(lo$t),
     label = paste0("dose #", unique(lo$dose)))
close.screen(all = TRUE)
par(op)

Gives:
 dose   t     C
    2 -96 40.08
    3 -72 51.87
    4 -48 55.77
    5 -24 56.64
    6   0 57.66

 dose   t     C
    2 -96 40.41
    3 -72 52.03
    4 -48 55.36
    5 -24 56.74
    6   0 57.25

 variability    int    slope signif # 0?     CL.lo     CL.hi
         low 57.635 0.039375       yes   +0.016450 +0.062300
        high 57.395 0.039375       no    -0.093589 +0.172339



─ low variability
─ high variability
dashed lines: linear regression

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• EMEA. Overview of Comments received on Draft Guideline on the Investigation of Bioequivalence. EMA/CHMP/EWP/26817/2010. London. 20 January 2010. Online.