## AUC · k for variable inter-occasion CL [NCA / SHAM]

Dear all,

let’s recap the basic mass balance equation of PK: $$\small{F\cdot D=V\cdot k\,\cdot\int_{0}^{\infty}C(t)\,dt=CL\cdot AUC_{0-\infty}}\tag{1}$$ We assess BE by comparative BA, i.e., $$\small{\frac{F_{\textrm{T}}}{F_{\textrm{R}}}\approx \frac{AUC_{\textrm{T}}}{AUC_{\textrm{R}}}}\tag{2}$$ That’s only part of the story because – based on $$\small{(1)}$$ – actually $$\small{AUC_{\textrm{T}}=\frac{F_\textrm{T}\cdot D_\textrm{T}}{CL}\;\land\;AUC_{\textrm{R}}=\frac{F_\textrm{R}\cdot D_\textrm{R}}{CL}}\tag{3}$$ Since (except for Health Canada and in exceptional cases for the EMA) a dose-correction is not acceptable, we have to assume that the true contents equal the declared ones and further $$\small{D_\textrm{T}\equiv D_\textrm{R}}\tag{4}$$ This allows us to eliminate the doses from $$\small{(3)}$$; however we still have to assume no inter-occasion variability of Clearances ($$\small{CL=\textrm{const}}$$) in order to arrive at $$\small{(2)}$$.
Great but is that true? If we have to deal with a Highly Variable Drug, the high variability is an intrinsic property of the drug itself (not the formulation). In BE were are interested in detecting potential differences of formulations, right? Since we ignored the – possibly unequal – Clearances, all unexplained variability goes straight into the resi­dual error, results in a high CV and hence, a wide confidence interval. In other words, the formulation is punished for a crime that Clearance has committed.
Can we do anything against it – apart from reference-scaling? We know that$$\small{k=CL\big{/}V}\tag{5}$$ In a cross-over design the volume of distribution of subjects likely shows limited inter-occasion variability. In a parallel design we can minimize its variability by carefully selecting subjects with similar anthropometric properties (body weight, age, etc.) in the groups. Therefore, we can drop the volume of distribution and approximate the ef­fect of $$\small{CL}$$ by $$\small{k}$$. This leads to $$\small{\frac{F_{\textrm{T}}}{F_{\textrm{R}}}\sim \frac{AUC_{\textrm{T}}\cdot k_{\textrm{T}}}{AUC_{\textrm{R}}\cdot k_{\textrm{R}}}}\tag{6}$$ A variant of $$\small{(6)}$$ – using $$\small{t_{1/2}}$$ instead of $$\small{k}$$ – was proposed already in the dark ages* by Wagner.1 Later work by an author of the FDA (!)2 was ignored as well. Hey, for 20+ years! A recent paper3 demonstrated its usefulness in extensive simulations.

Abstract
Aim: To quantify the utility of a terminal-phase adjusted area under the concentration curve method in increasing the probability of a correct and conclusive outcome of a bioequivalence (BE) trial for highly variable drugs when clearance (CL) varies more than the volume of distribution (V).
Methods: Data from a large population of subjects were generated with variability in CL and V, and used to simulate a two-period, two-sequence crossover BE trial. The 90% confidence interval for formulation comparison was determined following BE assessment using the area under the concentration curve (AUC) ratio test, and the proposed terminal-phase adjusted AUC ratio test. An outcome of bioequivalent, non-bioequivalent or inconclusive was then assigned according to predefined BE limits.
Results: When CL is more variable than V, the proposed approach would enhance the probability of correctly assigning bioequivalent or non-bioequivalent and reduce the risk of an inconclusive trial. For a hypothetical drug with between-subject variability of 35% for CL and 10% for V, when the true test-reference ratio of bioavailability is 1.15, a cross-over study of n=14 subjects analyzed by the proposed method would have 80% or 20% probability of claiming bioequivalent or non-bioequivalent, compared to 22%, 46% or 32% probability of claiming bioequivalent, non-bioequivalent or inconclusive using the standard AUC ratio test.
Conclusions: The terminal-phase adjusted AUC ratio test represents a simple and readily applicable approach to enhance the BE assessment of drug products when CL varies more than V.

I ❤️ the idea. When Abdallah’s paper2 was published, I tried it retrospectively in a couple of my studies. Worked mostly, and if not, it was a HVDP, where the variability is caused by the formulation (e.g., gastric-resistant diclo­fenac).

• When the portmanteau word ‘bioavailability’ (of ‘biological’ and ‘availability’) was in the future. Try to search for papers prior to ~1975 with ‘bioavailability’. You will be surprised.

1. Wagner JG. Method of Estimating Relative Absorption of a Drug in a Series of Clinical Studies in Which Blood Levels Are Measured After Single and/or Multiple Doses. J Pharm Sci. 1967; 56(5): 652–3. doi:10.1002/jps.2600560527.
2. Abdallah HY. An area correction method to reduce intrasubject variability in bioequivalence studies. J Pharm Pharm Sci. 1998; 1(2): 60–5. Open access.
3. Lucas AJ, Ogungbenro K, Yang S, Aarons L. Chen C. Evaluation of area under the concentration curve adjusted by the terminal-phase as a metric to reduce the impact of variability in bioequivalence testing. Br J Clin Pharmacol. Early View 16 July 2021. doi:10.1111/bcp.14986.

Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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