Bioequivalence and Bioavailability Forum

Hi Nastia!

❝ So what are the main requirements for the ideal PK metric?

You are asking a nasty question – rattling the foundations of BE by NCA. I like it. 

❝ Not to change and have low variability?

IMHO, it should be selective, i.e., reflect changes of what we are interested in. For the extent of absorption it’s clearly AUC_0–x. It’s another question what the x is… As we have seen in the other thread starting there, we get an unbiased estimate of T/R with AUC0_–∞ and AUC_{0–t(common)}. How early could we stop sampling (x = ?) to get an unbiased estimate is – as you know – on my to-do list. I still hold that for IR products it’s much earlier than regulators think.
For the rate of absorption? Well, cough. Since C_max is a composite metric, it can’t be selective for k_a. We don’t have to re-invent the wheel; reading educates.^1–6 Though I know >50% of the authors personally, I’m not biased. 
Interesting the abstract⁶

Results: The outcome of a bioequivalence trial was shown to depend on the measure. C_max/AUC reflected changes in k_a, but not in F. AUC showed dependence on F, but virtually no dependence on k_a. For C_max, a 3- to 4-fold increase in k_a and a concomitant 20% decrease in F, as well as corresponding changes in the opposite directions, resulted in bioequivalent outcomes.

But then

Conclusions: It was concluded that use of C_max/AUC should be discouraged and that defining bioequivalence in terms of rate and extent of absorption has major problems.

(my emphasis)
IMHO, the conclusion contradicts the results. How come? Only because two authors were from the FDA and they didn’t want to change the rules? Why am I not surprised?
A picture tells more than a thousand words. A funny one of the paper:

A C_max/AUC, B AUC, C C_max

With \(\small{\frac{k_{a,T}}{k_{a,R}}=5\,\wedge \frac{F_T}{F_R}=0.8}\) one has a high chance of passing C_max (given, will fail on AUC). OK but as stated in the abstract there are combinations where k_a and F are clearly different though both C_max and AUC will pass. Bravo, well done! Great metric.
Reminds me on another paper⁷ by authors of the FDA assessing the performance of AUC_0–t and AUC_0–∞. Results: T/R-ratios very similar, AUC_0–∞ more variable. The FDA’s consequences: Use both.

❝ Returning to C_max/AUC - do we need more strict conditions to prove BE on its base keeping in mind its much lower variability (narrow CI for example) - cause it's always within the limits?

Why? The conventional limits are based on the assumption that a ∆ of 20% is clinically not relevant. Leaving NTIDs aside, do we narrow the limits (say, for AUC) only because the variability of some drugs is low? Nope. If we would make the limits dependent on the variability, we would end up in reference-scaling chaos.

❝ (In old (e.g. below 2013) russian protocols I even saw the opposite situation when the confidence limits for C_max/AUC…

I know. Strange.

❝ … (as well as C_max) were choosen to be "75,00-133,00%")

I understand that. Before the European 2001 Note for Guidance, many products got even an approval with ∆ 30% (70.00–142.86%).

❝ Can you please also clarify the distribution of T_1/2: it is always presented as Mean±SD, is it correct? (Mean=Arithmetic Mean).

You rub salt into my wounds. Let’s step back. Rate constants have a unit of 1/time. Hence, the correct location parameter is the harmonic mean. Its dispersion parameter is the jackknife standard deviation (in WinNonlin’s terminology: Pseudo SD). For t_½ you have two options: Use the same as well (as I do though I’m not sure about the distribution; Γ?) or go with nonparametrics (x̃, quartiles).
Additional stuff at the end.

❝ It seemed to me I've read somewhere on the forum that for study planning we should use not the mean T_1/2 but the confidence level for it, but I couldn't find this thread now.

I’m too lazy to search as well. This one about extremes?

❝ Is it correct to use standard approach to calculate CI for untransformed T_1/2 as normally distributed data …

Anything is better than the mean (see this presentation, slides 64–66).

❝ … or should we use other approaches like nonparametric median confidence intervals or bootstrap?

Sounds good though “nonparametric CI” is a little bit strange. I think that I once saw a paper about it, not sure. However, if you don’t have data of a previous study… What you find in the public domain is often x±SD or min/max.

❝ ❝ E.g., for biphasic methylphenidate the cut-off time (FDA: 3 h fasting, 4 h fed) is based on PD indeed (at that time ~90% of patients show the maximum effect). […]

❝

❝ That's correct but too much individual approach - we couldn't know beforehand all the thin nuances of the PD of the specific drug, sometimes it is even impossible to find in literature any PK data for some drugs while planning new study... So I think the overall approach should be more general.

Early and late partial AUCs are only relevant for multiphasic MR products. Luckily I met only three so far: zolpidem, methylphenidate, amphetamine(s). For the first two there is a PD based justification possible. For the last one – no idea.

❝ Why not? The static moments of the plain curve 'Concentration-time' (C(t)) are:

❝

❝

\(M_t=\int t\cdot C(t)dt\);

❝ \(M_C=1/2\cdot\int C(t)^2dt\)

❝ First is connected with MRT (muliplyed by total square of the curve), and the second should reflect some ideal Concentration-unit parameter (the height of the gravity center). Why not to use it like some alternative to C_max?

Back in the days when I used my own software, I reported all it could calculate (hey, look how clever I am). Arrogant attitude. Confused sponsors and regulators. What I learned: The variability of VRT sucks. Not surprising cause we have \(\small{t^2}\) and \(\small{C^2}\) in it.

❝ ❝ Only nice to print a profile, cut it a out, push a pin through it, and make a weird whizz wheel for kids.

❝

❝ Oh, if we'll proceed further in the power of moments we should be able to calculate moment of inertia - then one can take that curve to the orbital station in order to research Dzhanibekov effect

Didn’t know that one! BTW, the center of gravity can be outside of the profile (scroll down in this post). Aboriginals know that for ages.

❝ ❝ To get a reliable estimate of AUC one has to cover 95% (!) of AUC_0–∞ (note that I’m not taking about BE but hard-core PK). For AUMC is should be 99%. I’m quoting Les Benet. Don’t blame me.

❝

❝ Could you please explain this more detaily? What do you mean in getting a reliable estimate of AUC?

I was talking about PK modeling. IIRC, Les Benet said that at the BioInternational in Munich 1994. Not sure whether it’s mentioned in the book.

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1. 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 Pharm Ther Toxicol. 1991;29(10):394–9. PMID 1748540.
   
2. Schall R, Luus HG. Comparison of absorption rates in bioequivalence studies of immediate release drug formulations. Int J Clin Pharmacol Ther. 1992;30(5):153–9. PMID 1592542.
   
3. 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.
   
4. Lacey LF, Keene ON, Duquesnoy C, Bye A. Evaluation of Different Indirect Measures of Rate of Drug Absorption in Comparative Pharmacokinetic Studies. J Pharm Sci. 1994;83(2):212–5. doi:10.1002/jps.2600830219.
   
5. Schall R, Luus HG, Steinijans VW, Hauschke D. Choice of Characteristics and Their Bioequivalence Ranges for the Comparison of Absorption Rates of Immediate-Release Drug Formulations. Int J Clin Pharmacol Ther. 1994;32(7):323–8. PMID 7952792.
   
6. Tozer TN, Bois FY, Hauck WW, Chen M-L, Williams RL. Absorption Rate Vs. Exposure: Which Is More Useful for Bioequivalence Testing? Pharm Res. 1996;13(3):453–56. doi:10.1023/a:1016061013606.
   
7. Bois FY, Tozer TN, Hauck WW, Chen M-L, Patnaik R, Williams RL. Bioequivalence: Performance of Several Measures of Extent of Absorption. Pharm Res. 1994;11(5):715–22. doi:10.1023/A:1018932430733.

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My post № 5,000.

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Edit: I explored my data. Drug X, 5–60 mg (linear PK proven), same bioanalytical method (enantio­selective stable isotope IS GC/MS, LLOQ dependent on the dose), sampling for 24 h, 3–7 time points for the estimation of λ_z, extrapolated AUC <10%.

I was wrong for many years. Seems that I have to revise my procedures and go with the median or geometric mean in the future.