Helmut ★★★ Vienna, Austria, 20210930 23:31 (112 d 22:33 ago) Posting: # 22608 Views: 1,152 

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 dosecorrection 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 interoccasion 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 referencescaling? We know that$$\small{k=CL\big{/}V}\tag{5}$$ In a crossover design the volume of distribution of subjects likely shows limited interoccasion 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 paper^{3} demonstrated its usefulness in extensive simulations. Abstract I ❤️ the idea. When Abdallah’s paper^{2} 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., gastricresistant diclofenac).
— Diftor heh smusma 🖖 Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
PharmCat ★ Russia, 20211001 00:15 (112 d 21:50 ago) @ Helmut Posting: # 22609 Views: 1,093 

Hey! Helmut, Sounds great. One problem ... I don't think that our regulators will be able to understand this, or there will be a desire and dare to change the foundations. Also, I think NLME modeling can be applied. 
Helmut ★★★ Vienna, Austria, 20211001 11:18 (112 d 10:46 ago) @ PharmCat Posting: # 22610 Views: 1,080 

Hi PharmCat, » I don't think that our regulators will be able to understand this, … Well, the majority of regulators is not stupid. That’s basic PK and \(\small{CL=\textrm{const}}\) is a pretty strong assumption, which might be outright false. Then the entire concept is built on sand: Studies are substantially larger than necessary, exposing innocent subjects to nasty drugs. » … or there will be a desire and dare to change the foundations. Correct. However, the FDA^{1} gives in its regulatory definitions of BE ‘rate and extent of absorption’ without mentioning any PK metrics, the EMA^{2} ‘BE with the reference medicinal product […] demonstrated by appropriate bioavailability studies’. Only the WHO^{3} gives specifically AUC for the extent of absorption. Too lazy to check other jurisdictions. Somehow it reminds me on the comparison of \(\small{AUC_{t_\textrm{last}}}\), which is known to be biased if \(\small{t_\textrm{last,T}\neq t_\textrm{last,R}}\). Nobody gives a shit. Remember the endless story of the inflated Type I Error in SABE? For HVDs (not HVDPs) probably we could counteract the high variability, use ABE with fixed limits, and all is good. Of course, it would mean changing the guidelines. Abdallah (FDA/CDER): It is recommended that area correction be attempted in bioequivalence studies of drugs where high intrasubject variability in clearance is known or suspected. […] The value of this approach in regulatory decision making remains to be determined. Lucas et al.:Performance of the AUC·k ratio test […] indicate that the regulators should consider the method for its potential utility in assessing HVDs and lessening unnecessary drug exposure in BE trials. » Also, I think NLME modeling can be applied. Hhm, can you elaborate?
P.S.: \(\small{C_\textrm{max}}\) is a lousy metric for comparing rates of absorption, since it is composite of \(\small{k_\textrm{a}}\) and \(\small{AUC_{0\infty}}\). Lots of publications… What about: $$\small{\frac{F_\textrm{T}}{F_\textrm{R}}\sim \frac{C_\textrm{max,T}\big{/}\left(AUC_{0\infty,\textrm{T}}\cdot k_\textrm{T} \right)}{C_\textrm{max,R}\big{/}\left(AUC_{0\infty,\textrm{R}}\cdot k_\textrm{R} \right)}}$$ Deserves a paper. — Diftor heh smusma 🖖 Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
d_labes ★★★ Berlin, Germany, 20211001 13:52 (112 d 08:12 ago) @ Helmut Posting: # 22611 Views: 947 

Dear both, what do you think is a reasonable assumption about the distribution of the metric AUC*k? Do we have to throw away our evaluation of BE studies assuming lognormal distri of the metrics AUC and/or Cmax? — Regards, Detlew 
Helmut ★★★ Vienna, Austria, 20211001 17:01 (112 d 05:04 ago) @ d_labes Posting: # 22612 Views: 962 

Dear Detlew, » what do you think is a reasonable assumption about the distribution of the metric AUC*k? Since both are lognormal, their ratio should be lognormal as well. I trust here Martin; will meet him in the evening and ask again. Furthermore, the distribution of values per se is not important, only the residual error. One of my 4period full replicate studies (143 subjects, Method A) » Do we have to throw away our evaluation of BE studies assuming lognormal distri of the metrics AUC and/or Cmax? Not at all – if they passed. Too bad if they failed and would have passed with AUC·k… — Diftor heh smusma 🖖 Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
d_labes ★★★ Berlin, Germany, 20211002 10:55 (111 d 11:10 ago) @ Helmut Posting: # 22614 Views: 853 

Dear Helmut, » Since both are lognormal, their ratio should be lognormal as well. I trust here Martin; will meet him in the evening and ask again. » Furthermore, the distribution of values per se is not important, only the residual error. Correct. » One of my 4period full replicate studies (143 subjects, Method A) » Looks not too bad for a lognormal . — Regards, Detlew 
Helmut ★★★ Vienna, Austria, 20211003 20:29 (110 d 01:36 ago) @ d_labes Posting: # 22616 Views: 832 

Dear Detlew, » » » what do you think is a reasonable assumption about the distribution of the metric AUC*k? » » Since both are lognormal, their ratio should be lognormal as well. I trust here Martin; will meet him in the evening and ask again. Answer: The difference of two normal distributions is normal distributed. The ratio and product of two lognormal distributions are lognormal distributed. Furthermore, the unit of AUC·k is mass/volume.
— Diftor heh smusma 🖖 Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
Helmut ★★★ Vienna, Austria, 20211004 12:05 (109 d 09:59 ago) @ d_labes Posting: # 22618 Views: 808 

Dear Detlew, » » One of my 4period full replicate studies (143 subjects, Method A) » » Looks not too bad for a lognormal . More details… The reference formulation in this study was terrible; CV_{wR} twice of CV_{wT}, many subjects with low AUCs after R. Distributions of studentized model residuals heavytailed. Since CV_{wT} <30%, this was a HVDP and not a HVD. Adjusting the AUC would make things worse. 90% CIs (Method B, Satterthwaite’s df): AUC_{0–∞}: 111.17 – 123.79% AUC_{0–∞}·k: 113.89 – 132.70%
— Diftor heh smusma 🖖 Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
mittyri ★★ Russia, 20211004 14:22 (109 d 07:42 ago) @ Helmut Posting: # 22619 Views: 775 

Dear Helmut, » More details… The reference formulation in this study was terrible; CV_{wR} twice of CV_{wT}, many subjects with low AUCs after R. Since CV_{wT} <30%, this was a HVDP and not a HVD. Adjusting the AUC would make things worse. now try something with horrible enterohepatic cirulation — Kind regards, Mittyri 
Helmut ★★★ Vienna, Austria, 20211004 16:10 (109 d 05:54 ago) @ mittyri Posting: # 22620 Views: 731 

Hi mittyri, » now try something with horrible enterohepatic cirulation The masochistic part of my psyche is limited.* However, the method is not a Swiss armyknife. I would never perform a pivotal replicate study without a pilot in order to see what’s going on.
— Diftor heh smusma 🖖 Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
PharmCat ★ Russia, 20211001 18:48 (112 d 03:17 ago) @ Helmut Posting: # 22613 Views: 966 

» Hhm, can you elaborate? these are my thoughts, I did not support them with any deep search ... $$C = C_0 * e^{k_e * t}$$ => $$C = \frac{D}{V_d} * e^{k_e * t}$$ Model: $$C = \frac{aX + \epsilon_2}{V_d} * e^{bZ * t} * \epsilon_1 $$ $$ \epsilon_1 \sim N(0, \sigma_1^2); \epsilon_2 \sim N(0, \sigma_2^2)$$ It can be solved for \(a, V_d, b, \sigma_1^2, \sigma_2^2\) and modified for extravascular model. Something described here: Korkmaz, Selcuk & Orman, Mehmet. (2012). The Use of Nonlinear Mixed Effects Models in Bioequivalence Studies: A Real Data Application. Open Access Scientific Reports. 111 However, I think terminalphase adjusted area under the concentration curve method is something between the classic and NLME approaches. And both of them make corrections for variation from elimination. 