Bioequivalence and Bioavailability Forum

Hi all,

I have several questions regarding the simulation of the inter-subject variability in the 2x2x4 replicate study. It's a long post, so let's use concrete example so it's easier to discuss.

Imagine I need to simulate a 2x2x4 study with n = 24 subjects (so 12 for TRTR and 12 for RTRT). The mean Cmax of test and reference is 100 ng/mL, intra-subject variability for test is sd.t = 0.2 and for reference is sd.r = 0.1. Here the variability is expressed as standard deviation (SD), not CV.
cmax.r <- rnorm(48, mean = log(100), sd = 0.1)
cmax.t <- rnorm(48, mean = log(100), sd = 0.2)
This will simulate 48 Cmax values for T and R. With a bit coding, we can generate TRTR/RTRT sequences and randomly assign 24 subjects to those sequences with corresponding period and Cmax values. We can back-calculate the within-subject variability of T and R either using EMA's method (e.g., using only reference data for within-subject variability of reference as described in Clinical pharmacology and pharmacokinetics: questions and answers.) or FDA's method (e.g., described in the appendix C of the BE guidance). Easy enough, so far so good.

Q1: In this case, based on those population parameters described above, what is the expected inter-subject variability? Is there mathematical formula to get the between-subject SD of T and R? or pooled SD?

Q2: If there's no mathematical derivation for between-subject SD, I think that we can estimate them using R. But how?

One way that I am thinking is to use EMA's method, removing test product from the data set and do ANOVA as described in EMA's Q&A.
m.ref <- anova(lm(log(cmax) ~ sequence + subject %in% sequence + period, data = d.ref))
The code above should give us the within and between-subject SD of reference with swr = sqrt(MSResidual) and sbr = sqrt((MSSubject(seq) - MSResidual)/2), where swr and sbr are within-and between-SD of reference, respectively. Similarly we can do it for the test. And in case MSSubject(seq) < MSResidual, we can try what Detlew described in another post to avoid negative variance.

Q3: Is this a correct approach? If not, what's the correct one?

Now comes to simulation. Imagine in addition to what I described above for simulating Cmax of T and R with mean value of 100 ng/mL, sd.t = 0.2 and sd.r = 0.1, I also want to introduce between-subject SD for T and R, say sd.bt = 0.4 and sd.br = 0.2, where sd.bt and sd.br are between-subject SD for T and R.

Q4: How do we do it?

I saw that Helmut did it as follows in one of his articles (Helmut, it's Simulations 101 but the link causes Error 406 .) for simulating the 2x2x2 crossover study.
T  <- rnorm(n = n, mean = log(theta0), sd = sd)
R  <- rnorm(n = n, mean = 0, sd = sd)
TR <- rnorm(n = n, mean = 0, sd = sd.b)
T  <- T + TR
R  <- R + TR
where sd.b is the between-subject SD if I understood it correctly.

However, when I tried to do it similarly for 2x2x4 as follows and back-calculate the within and between SD, the results doesn't match expectation at all because the between-subject SD fromANOVA is extremely small, instead of around the expected values of 0.2 or 0.4.
cmax.r <- rnorm(48, mean = log(100), sd = 0.1)
cmax.t <- rnorm(48, mean = log(100), sd = 0.2)
cmax.br <- rnorm(48, mean = log(100), sd = 0.2) # for reference
cmax.bt <- rnorm(48, mean = log(100), sd = 0.4) # for test
# I need the concentration in original scale, not log-transformed
cmax.r <- exp(cmax.r + cmax.br)
cmax.t <- exp(cmax.t + cmax.bt)

Q5: What went wrong? Should the mean value for between-subject SD of T and R always 0 instead of using the same value as T and R?

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Edit: Category changed. [Helmut]