Bioequivalence and Bioavailability Forum

Hi libaiyi,

❝ When we get a CVwr, how to get the BE limits?

The switching CV_wR is 30% and hence, the regulatory constant k should be \(\log{(1.25)}/\sqrt{\log{(0.30^2+1)}} = 0.7601283\ldots \)
In R (CVs and BE limits in %):

CVwR <- seq(0.25, 0.55, 0.05)
swR  <- sqrt(log(CVwR^2+1))
k    <- log(1.25)/sqrt(log(0.3^2+1))
EL1  <- data.frame(CVwR=100*CVwR, swR=swR, L=100*exp(-k*swR), U=100*exp(+k*swR))
EL1$L[EL1$CV <= 30] <-  80 # conventional
EL1$U[EL1$CV <= 30] <- 125 # BE limits
EL1$L[EL1$CV >= 50] <- 100*exp(-k*sqrt(log(0.5^2+1))) # upper scaling
EL1$U[EL1$CV >= 50] <- 100*exp(+k*sqrt(log(0.5^2+1))) # cap at CV 50%
print(EL1, row.names=FALSE)

CVwR       swR        L        U
  25 0.2462207 80.00000 125.0000
  30 0.2935604 80.00000 125.0000
  35 0.3399387 77.22885 129.4853
  40 0.3852532 74.61401 134.0231
  45 0.4294214 72.15055 138.5991
  50 0.4723807 69.83255 143.1997
  55 0.5140870 69.83255 143.1997

In its eternal wisdom (preferring “nice” numbers) the EMA demands a rounded k 0.76 (which implies a switching CV_wR of 30.0053%).*

EL2 <- data.frame(CVwR=100*CVwR, swR=swR, L=100*exp(-0.76*swR), U=100*exp(+0.76*swR))
EL2$L[EL2$CV <= 30] <-  80
EL2$U[EL2$CV <= 30] <- 125
EL2$L[EL2$CV >= 50] <- 100*exp(-0.76*sqrt(log(0.5^2+1)))
EL2$U[EL2$CV >= 50] <- 100*exp(+0.76*sqrt(log(0.5^2+1)))
print(EL2, row.names=FALSE)

CVwR       swR        L        U
  25 0.2462207 80.00000 125.0000
  30 0.2935604 80.00000 125.0000
  35 0.3399387 77.23222 129.4796
  40 0.3852532 74.61770 134.0165
  45 0.4294214 72.15452 138.5915
  50 0.4723807 69.83678 143.1910
  55 0.5140870 69.83678 143.1910

More convenient by the function scABEL() of the R-package PowerTOST (by applying the default argument regulator="EMA"). The function then uses k 0.76, returns the conventional ABE limits if CV_wR ≤30%, and observes the upper cap of CV_wR 50%:

library(PowerTOST)
CVwR <- seq(0.25, 0.55, 0.05)
EL3  <- data.frame(CVwR=100*CVwR, swR=CV2se(CVwR),
                   L=100*scABEL(CV=CVwR)[, "lower"],
                   U=100*scABEL(CV=CVwR)[, "upper"])
print(EL3, row.names=FALSE)

CVwR       swR        L        U
  25 0.2462207 80.00000 125.0000
  30 0.2935604 80.00000 125.0000
  35 0.3399387 77.23222 129.4796
  40 0.3852532 74.61770 134.0165
  45 0.4294214 72.15452 138.5915
  50 0.4723807 69.83678 143.1910
  55 0.5140870 69.83678 143.1910

Don’t apply double rounding! Round the 90% CI in percent to two decimal places but compare it to the expanded limits in full precision.

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• Karalis V, Symillides, Macheras P. On the leveling-off properties of the new bioequivalence limits for highly variable drugs of the EMA guideline. Eur J Pharm Sci. 2011;44:497–505. doi:10.1016/j.ejps.2011.09.008.

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Edit: Hey Shuanghe, you were faster!